# Discrete Geometric Singular Perturbation Theory

S. Jelbart & C. Kuehn∗ Department of Mathematics, The Technical University of
Munich. Garching Bavaria, 85748, Germany.

###### Abstract

We propose a mathematical formalism for discrete multi-scale dynamical systems
induced by maps which parallels the established geometric singular
perturbation theory for continuous-time fast-slow systems. We identify
limiting maps corresponding to both ‘fast’ and ‘slow’ iteration under the map.
A notion of normal hyperbolicity is defined by a spectral gap requirement for
the multipliers of the fast limiting map along a critical fixed-point manifold
$S$. We provide a set of Fenichel-like perturbation theorems by reformulating
pre-existing results so that they apply near compact, normally hyperbolic
submanifolds of $S$. The persistence of the critical manifold $S$, local
stable/unstable manifolds $W^{s/u}_{loc}(S)$ and foliations of
$W^{s/u}_{loc}(S)$ by stable/unstable fibers is described in detail. The
practical utility of the resulting discrete geometric singular perturbation
theory (DGSPT) is demonstrated in applications. First, we use DGSPT to
identify singular geometry corresponding to excitability, relaxation, chaotic
and non-chaotic bursting in a map-based neural model. Second, we derive
results which relate the geometry and dynamics of fast-slow ODEs with non-
trivial time-scale separation and their Euler-discretized counterpart.
Finally, we show that fast-slow ODE systems with fast rotation give rise to
fast-slow Poincaré maps, the geometry and dynamics of which can be described
in detail using DGSPT.

Keywords: geometric singular perturbation theory, discrete dynamical systems,
multi-scale dynamical systems, singularly perturbed maps, invariant manifolds.

MSC2020: 37C05, 37D10, 37C86, 34D15, 37C15.

##  1 Introduction

The primary aim of this manuscript is to provide a mathematical framework for
the geometric analysis of multi-scale discrete dynamical systems induced by
maps. In essence, we aim to provide a discrete geometric singular perturbation
theory, or simply (DGSPT), which parallels the established geometric singular
perturbation theory (GSPT) for continuous-time fast-slow systems [17, 34, 36,
42, 72]. Our starting point is motivated by the recently developed formulation
in [72] of GSPT for continuous-time, $C^{r}-$smooth fast-slow ODE systems in
the general form

$z^{\prime}=N(z)f(z)+\varepsilon G(z,\varepsilon),$ (1)

where $z\in\mathbb{R}^{n}$, $z^{\prime}=dz/dt$, $N$ is an $n\times(n-k)$
matrix, $f$ and $G$ are vector-valued functions of dimensions $(n-k)\times 1$
and $n\times 1$ respectively, $0<\varepsilon\ll 1$ is a perturbation parameter
and the existence of a $(k<n)-$dimensional critical manifold
$S=\\{z\in\mathbb{R}^{n}:f(z)=0\\}$ for $\varepsilon=0$ is assumed. The class
of systems defined by (1) includes the (perhaps better known) class of fast-
slow systems in the so-called standard form

$\begin{split}x^{\prime}&=\varepsilon\tilde{g}(x,y,\varepsilon),\\\
y^{\prime}&=\tilde{f}(x,y,\varepsilon),\end{split}$ (2)

with $(x,y)\in\mathbb{R}^{k}\times\mathbb{R}^{n-k}$ as a special case, since
(2) can be written in the form (1) after setting $z=(x,y)^{\top}$,
$f(z)=\tilde{f}(x,y,0)$, $N(z)=(O_{k,n-k},I_{n-k})^{\textnormal{T}}$ and
$G(z,\varepsilon)=(\tilde{g}(x,y,\varepsilon),\tilde{f}(x,y,\varepsilon)-\tilde{f}(x,y,0))^{\textnormal{T}}$.
Conceptually, the formulation presented in [72] can be considered as a
coordinate-independent extension of earlier formulations for standard form
systems (2) in e.g. [34, 36, 42], motivated by a large number of applications
for which fast-slow dynamics arises in systems that are ‘beyond the standard
form’, see e.g. [7, 25, 31, 32, 35, 37, 38, 39, 44, 61].

For the development of DGSPT, we consider fast-slow maps in the general form

$z\mapsto\bar{z}=z+N(z)f(z)+\varepsilon G(z,\varepsilon),$ (3)

where $z\in\mathbb{R}^{n}$, $N$, $f$ and $G$ are defined as in the continuous-
time system (1), $0<\varepsilon\ll 1$ is a perturbation parameter and the
existence of a $(k<n)-$dimensional critical (fixed point) manifold
$S=\\{z\in\mathbb{R}^{n}:f(z)=0\\}$ for $\varepsilon=0$ is assumed. Similarly
to continuous-time setting, the class of maps defined by (3) includes the
(perhaps better known) class of fast-slow maps in standard form

$\begin{split}x&\mapsto\bar{x}=x+\varepsilon\tilde{g}(x,y,\varepsilon),\\\
y&\mapsto\bar{y}=y+\tilde{f}(x,y,\varepsilon),\end{split}$ (4)

where $(x,y)\in\mathbb{R}^{k}\times\mathbb{R}^{n-k}$ as a special case, since
(4) can be written in the form (3) using with the same choices for $N$, $f$
and $G$ which put system (2) into the form (1).

The utility of GSPT in the continuous-time setting depends upon the
availability of both

  * (I)

Singular theory: A mathematical framework or formalism for the geometric
analysis of non-equivalent limiting problems for each ‘time-scale’ as
$\varepsilon\to 0$, and

  * (II)

Perturbation/invariant manifold theorems: A collection of results on the
perturbation of dynamical and geometric structure identified using the
singular theory in (I) for $0<\varepsilon\ll 1$,

and this work, we shall consider (I) and (II) together as the basic
requirements for a prospective DGSPT.

In the continuous-time setting, a singular theory as required by (I) was
already present in [17], later refined and clarified for fast-slow systems in
standard form (2) in e.g. [21, 34, 36, 42, 65], and finally for the more
general class (1) in [72] and related works, e.g. [8, 20, 40, 47, 48, 49].
This theory provides a mathematical framework for identifying and analysing
the geometry and dynamics of the so-called layer (or fast subsystem) and
reduced (or slow subsystem) problems, obtained after taking $\varepsilon\to 0$
in (1) on the fast and slow time-scales $t$ and $\tau=\varepsilon t$
respectively. A typical analysis proceeds by constructing ‘singular orbits’,
sometimes also called candidate orbits, by a geometric concatenation of fast
trajectory segments of the layer problem on $\mathbb{R}^{n}\setminus S$ and
slow trajectory segments of the reduced problem on $S$.

The fundamental perturbation theorems required by (II) are provided by
Fenichel theory, which is constituted by a collection of perturbation and
invariant manifold theorems derived in [15, 16, 18] and culminating in [17]
(though it is also important to mention the earlier works of [24, 67], which
provided foundational understanding for the field). Fenichel theory ensures
that suitably constructed singular orbits perturb in a regular fashion for
$0<\varepsilon\ll 1$ in regions of phase space bounded away from certain
singularities on $S$. Fenichel theory applies locally near normally hyperbolic
submanifolds of $S$ for which the linearization along the ($k-$dimensional)
critical manifold $S$ has $n-k$ non-trivial eigenvalues bounded away from the
imaginary axis. For fast-slow systems (1), this is equivalent to the
requirement that the $(n-k)\times(n-k)$ square matrix $DfN|_{S}$ are bounded
away from the imaginary axis [72]. Under normally hyperbolic conditions, the
geometry and dynamics for $0<\varepsilon\ll 1$ are described up to
$O(\varepsilon)$ accuracy by the dynamics of layer and reduced problems, which
are almost invariably far more tractable analytically. Additional perturbation
theorems are required in order to describe the geometry and dynamics near non-
normally hyperbolic submanifolds of $S$. Many authors have demonstrated the
utility of a method of geometric desingularization known as the blow-up method
for such purposes; here we simply cite the seminal works in [11, 41] and refer
to the recent survey [29]. This combination of GSPT and blow-up techniques has
been applied by many authors [4, 9, 23, 25, 30, 31, 37, 38, 39, 44, 66].

In the discrete setting, the picture is less complete. To the best of our
knowledge, a singular theory in the sense of (I) does not yet exist for maps.
This is perhaps because of difficulties relating to the fact that there is no
direct analogue for the time rescaling $\tau=\varepsilon t$ which leads to an
equivalent ‘slow formulation’ of the map (3). Thus it is not immediately clear
how a to obtain a ‘reduced map’ which describes slow iteration on or close to
$S$. On the other hand, the perturbation and invariant manifold theory
required by (II) for the (discrete analogue of the) normally hyperbolic regime
is in principle quite established, dating back at least to the work of Hirsch,
Pugh & Shub [24], and results on the existence of invariant manifolds and the
foliation of the adjacent space in 2-dimensional maps arising in the analysis
of fast-slow ODE systems in particular are given in [66]. We also mention
Pötzsche [57], Nipp & Stoffer [51, 54] and Shub [63] (many more references can
be found in the books [54, 63]). In many practical situations, however, a
direct application of the pre-existing results of the results in [24] can be
difficult due to their generality and the relatively abstract formulation of
necessary and sufficient conditions of their applicability. On the other hand,
concrete and more applicable formulations such as those in Nipp & Stoffer [54]
depend upon the identification of suitable coordinates, and often require the
rather extensive use of nonlinear coordinate transformations and cutoff
techniques in order to ‘prepare’ the equations. Thus in many situations, there
remains a practical barrier to the application of these results. With regard
to perturbation results on the dynamics in the non-normally hyperbolic setting
we mention [1, 2, 10, 12, 13, 14, 19, 54, 55].

In order to obtain a satisfactory DGSPT, our first task is to develop a
singular theory in the sense of (I) for fast-slow maps (3). We begin by
defining a layer map by setting $\varepsilon=0$ in (3), which allows us
introduce a notion of normal hyperbolicity of $S$ in terms of a spectral gap
requirement. Algebraic formulae for the non-trivial multipliers needed to
determine the normal hyperbolicity of $S$ are given solely in terms of the
initial data $N$ and $f$. Specifically, it suffices to verify the existence of
an annular spectral gap about the unit circle for the matrix
$I_{n-k}+DfN|_{S}$.

Next, we show that a reduced map which approximates slow iteration near $S$ to
an accuracy of $O(\varepsilon^{2})$ can be derived under normally hyperbolic
conditions. The reduced map is conceptually distinct from the reduced problem
in the continuous-time setting since it reduces to the trivial map $z\mapsto
z$ as $\varepsilon\to 0$. This is necessarily so, since the
$\varepsilon-$dependence in the leading order cannot in general be ‘divided
out’ (as is achieved by moving to the slow time-scale $\tau=\varepsilon t$ in
the continuous-time setting). Fortunately, however, this is no obstacle in
practice, where one is primarily concerned with understanding the leading
order dynamics near $S$. We also show that a reduced $m$’th iterate map
induced by repeated iteration of (3) can be derived locally near $S$ using the
asymptotic self-similarity properties of (3). Interestingly, this $m$’th
iterate map can be related to a suitable discretization of the continuous-time
reduced problem associated to fast-slow ODEs (1) if the number of iterates $m$
is comparable to $\varepsilon^{-1}$. From an applied point of view, it is
significant that both reduced and $m$’th iterate maps are given by closed form
algebraic formulae defined purely in terms of $N$, $f$ and $G$.

Given the prevalence of fast-slow maps in standard form (4) in applications,
we also consider the form of the corresponding singular theory as a special
case of the theory developed for general fast-slow maps (3). In addition to
describing general features of the layer and reduced maps, we prove local
equivalence of the maps (3) and (4) near an arbitrary point on $S$. Just as in
the continuous-time setting, however, it is worthy to emphasise such an
equivalence is strictly local, and typically only useful for theoretical
purposes.

Having developed a singular theory, we turn our attention to the coupling of
this singular theory to suitable perturbation and invariant manifold theorems
as required by (II). This is achieved via the adaptation of pre-existing
perturbation and invariant manifold theorems in the formulation of [54] for
fast-slow maps (3). We provide persistence theorems which parallel Fenichel’s
invariant manifold theorem’s for flows in continuous-time fast-slow systems,
which apply under normally hyperbolic conditions in the discrete sense
described above. These results characterise the perturbation of compact
normally hyperbolic submanifolds of the critical manifold $S$, as well as its
corresponding local stable and unstable manifolds $W_{loc}^{s}(S)$ and
$W_{loc}^{u}(S)$ respectively for $0<\varepsilon\ll 1$. Local invariance and
smoothness properties of the perturbed counterparts are described, as well as
the asymptotic rate foliation of perturbed stable and unstable manifolds by
stable and unstable fast fibers, respectively.

Although most of the invariant manifold theorems presented herein have direct
and in most cases more general analogues in [54] and other pre-existing work
dating back to [24], our main contribution is to extend and reformulate these
results in a manner well-suited to applications. We emphasise in particular
the following:

  * •

As they are formulated herein, perturbation theorems do not depend on a
special choice of coordinates, so the equations do not need to be ‘prepared’.

  * •

In applications it suffices to check normal hyperbolicity of $S$, which
amounts to calculating the eigenvalues of the matrix $DfN|_{S}$.

  * •

Results apply to compact, normally hyperbolic submanifolds $S_{n}\subseteq S$.
Perturbed counterparts of $S$, $W^{s/u}_{loc}(S)$ and foliations for
$W^{s/u}_{loc}(S)$ are typically locally (as opposed to globally) invariant
objects.

In essence, our formulation leads to perturbation theorems which parallel
Fenichel’s theorems in the continuous-time setting. Our main perturbation
theorems are obtained by formulating a number of necessary and sufficient
conditions in [54] in terms of spectral bounds for the multipliers of the
layer map along $S$, which do not depend on a special choice of coordinates.
This allows us to derive results for the map (3) via the application of
results in [54] to a suitable ‘normal form’. The ‘price’ of coordinate-
independence in our approach, is that it requires a sufficient degree of
smoothness in the map (3) (it must be at least $C^{1}$ in order for the
spectrum to be well-defined), while many results in [54] apply minimally under
Lipschitz continuous conditions. The extension to the locally (as opposed to
globally) invariant case is delicate but standard, and achieved by the use of
cutoff techniques.

In addition to the reformulation of results in [54] as coordinate-independent
results in the locally invariant setting, we also provide concrete results for
the map (3), including explicit formulae for the perturbed slow manifold
$S_{\varepsilon}$ in local coordinates up to $O(\varepsilon^{2})$, and
quantitative estimates for contraction and repulsion along stable and unstable
fibers respectively in terms of the size of the spectral gap associated to the
matrix $I_{n-k}+DfN|_{S}$.

The utility of DGSPT is demonstrated in the context of three different
applications. The first of these is a 2-dimensional map-based model for
neuronal bursting known as the Chialvo map, introduced in [5] and considered
further in e.g. [33, 68]. This map takes the standard form (4) in a suitable
parameter regime, with an S-shaped critical manifold having two regular fold
points and a flip-type (period-doubling) bifurcation in the layer map
appearing as non-normally hyperbolic singularities on $S$. We show how DGSPT
can be used to identify four open parameter sets corresponding to excitable,
relaxation-type, non-chaotic bursting and (potentially chaotic) bursting
dynamics. This analysis extends (in some directions) the work in [5, 33, 68],
though it remains only partial since the dynamics near the non-normally
hyperbolic fold points is not yet understood in detail.

The last two applications demonstrate the utility of DGSPT in a more
theoretical setting. In the first of these, we consider the geometry and
dynamics of maps arising by Euler discretization of fast-slow systems in the
general non-standard form (1). These results, which relate the geometry and
dynamics of the ODE and discretized systems, parallel pre-existing results on
discretized fast-slow systems for larger classes of discretizations in e.g.
[52, 53]. We restrict to the simplest case of Euler discretized systems in
order to show clearly how DGSPT applies in such contexts. To the best of our
knowledge, the extension of these results to case of fast-slow systems in the
more general form (3) is also novel.

Finally, we use DGSPT in order to analyse fast-slow Poincaré maps associated
to fast-slow systems in standard with a single slow variable, a situation
which arises often in applications if one allows a parameter $\alpha$ to
evolve slowly in time; see e.g. [3, 19] and the references therein for
examples in the context of dynamic bifurcation theory. On the assumption that
the (continuous-time) layer problem has a hyperbolic limit cycle for
$\alpha=\alpha_{\ast}$, there exists a 2-dimensional manifold of limit cycles
$\mathcal{M}$ in $\mathbb{R}^{n}\times V_{\alpha}$, where $V_{\alpha}$ is a
sufficiently small neighbourhood about $\alpha_{\ast}$ (see e.g. [64] for an
example in 3 dimensions). After showing that the Poincaré map on a transverse
section $\Delta$ is a fast-slow map with normally hyperbolic critical manifold
$S=\Delta\cap\mathcal{M}$, we are able to characterise the geometry and
dynamics of the Poincaré map in detail using DGSPT. Using information about
the Poincaré map we can then infer geometric and dynamical properties for the
higher-dimensional ODE system, including the persistence of $\mathcal{M}$ a
nearby locally invariant manifold $\mathcal{M}_{\varepsilon}$.

The manuscript is structured as follows: In Section 2 we develop the singular
theory. Layer and reduced maps are introduced and characterised in Sections
2.1 and 2.2 respectively, and the relationship to the special subclass of
fast-slow maps in standard form (4) is considered in Section 2.3. The main
invariant manifold theorems are stated and described in Section 3, and proved
in Section 4. The applications are treated in Section 5; Section 5.1 contains
the geometric analysis of the map-based neural model, Section 5.2 contains the
analysis of Euler discretized fast-slow ODEs, and the application to fast-slow
Poincaré maps induced by ODEs with slowly varying parameters and persistence
results for limit cycle manifolds are given in Section 5.3. Finally in Section
6, we summarise and conclude the manuscript.

##  2 Coordinate-independent GSPT for fast-slow maps

In this section we extend the singular GSPT framework developed for fast-slow
ODEs in [72] to fast-slow maps. Specifically, we consider maps of the form

$z\mapsto\bar{z}=H(z,\varepsilon),$ (5)

with variables $z\in\mathbb{R}^{n}$ and perturbation parameter $\varepsilon$.
The map $H:\mathbb{R}^{n}\times[0,\varepsilon_{0})\to\mathbb{R}^{n}$ is
assumed to be $C^{r}-$smooth in both $z$ and $\varepsilon$. For simplicity, we
shall assume that $r$ is sufficiently large for the validity of certain
calculations. This will frequently lead to simplified statements, however most
results can be derived and stated minimally for $r\geq 1$.

Since (5) is (at least) $C^{1}-$smooth in $\varepsilon$, the map can be
written as

$z\mapsto\bar{z}=z+h(z)+\varepsilon G(z,\varepsilon),$ (6)

where the functions $h:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and
$G:\mathbb{R}^{n}\times[0,\varepsilon_{0})\to\mathbb{R}^{n}$ are
$C^{r}-$smooth in $z$ and (for the latter) $C^{r-1}-$smooth in $\varepsilon$.

######  Definition 2.1.

(Regularly perturbed maps/fast-slow maps) The map (6) is called a fast-slow
map if the level set

$C:=\left\\{z\in\mathbb{R}^{n}:h(z)=O_{n}\right\\},$ (7)

where $O_{n}$ denotes the $n\times 1$ zero vector, contains a
$(k<n)-$dimensional regularly embedded submanifold $S\subset\mathbb{R}^{n}$.
We also say that the map (6) is singularly perturbed. If the above condition
is not satisfied, we say that the map (6) is regularly perturbed.

Definition 2.1 is directly analogous to the geometric definition of singularly
perturbed ODEs in [17], see also [72, Definitions 3.1-3.2]. In this work we
are interested in the dynamics of fast-slow maps, so we restrict our focus
accordingly.

######  Assumption 1.

(Restriction to fast-slow maps) The map (6) is a fast-slow map in the sense of
Definition 2.1. We also assume for simplicity that the submanifold $S\subseteq
C\subset\mathbb{R}^{n}$ in Definition 2.1 is connected.

As stated, the assumption that $S$ is connected is made for simplicity.
Generalisations and adaptions of all results to cases in which $S$ is a
disjoint union of connected submanifolds are straightforward.

###  2.1 The layer map

Since (5) is smooth (and thus continuous) in $\varepsilon$, the limit
$\varepsilon\to 0$ is well-defined.

######  Definition 2.2.

(Layer map) The map

$z\mapsto\bar{z}=z+h(z)$ (8)

obtained from (10) in the limit $\varepsilon\to 0$ is called the layer map.

It follows from Assumption 1 that the set of fixed points $C$ contains a
$k-$dimensional submanifold $S$. In order that this submanifold $S$ itself has
a level set representation, we impose an algebraic assumption on the existence
of a suitable factorisation of $h(z)$. Such assumptions are also made in the
ODE setting in [72].

######  Assumption 2.

(Factorisation of the layer map) The function $h(z)$ can be factorised as
follows:

$h(z)=N(z)f(z),$ (9)

where $N(z)$ and $f(z)$ are matrices of size $n\times(n-k)$ and $(n-k)\times
1$ respectively. We assume the matrix $N(z)$ has full column rank for all
$z\in S$, and that fixed points $z_{\ast}\notin S$ such that
$N(z_{\ast})f(z_{\ast})=O_{n}$, if they exist, are isolated.

Using Assumption 2, we shall hereafter write the map (6) in the form

$z\mapsto\bar{z}=z+N(z)f(z)+\varepsilon G(z,\varepsilon).$ (10)

All subsequent results will be stated for fast-slow maps in the form (10)
satisfying Assumptions 1-2.

######  Remark 2.3.

The factorisation (9) can always be locally attained. In fact, algebraic
algorithms for obtaining such a local factorisation already exist [20, 49].
Globally, however, it must be assumed. See [72, Rem. 3.6] for discussion and
[7, Sec. 1.3] for counterexamples in the continuous-time setting.

It follows from Assumption 2 that the $k-$dimensional manifold of fixed points
$S\subseteq C$ is given by the level set

$S:=\left\\{z\in\mathbb{R}^{n}:f(z)=O_{n-k}\right\\}.$ (11)

In the theory for fast-slow ODEs, $S$ is known as the critical manifold. We
adopt the same terminology order to emphasise the similar role played by the
fixed point manifold (11) in fast-slow maps.

######  Definition 2.4.

(Critical manifold) The fixed point manifold $S$ is called the critical
manifold of (10).

Since $S$ is a regularly embedded submanifold of $\mathbb{R}^{n}$ by
Assumption 1, the $(n-k)\times n$ matrix $Df|_{S}$ is regular, i.e. has full
row rank.

######  Remark 2.5.

It is common in applications that the set $S$ is not everywhere a regularly
embedded submanifold in $\mathbb{R}^{n}$, since it contains, e.g. self-
intersections. We do not consider such cases in this work; they are ruled out
by Assumption 1.

In order to describe stability properties of the critical manifold, we
calculate the Jacobian

$DH(z,0)=I_{n}+Dh(z)=I_{n}+N(z)Df(z),\qquad z\in S,$ (12)

where $I_{n}$ denotes the $n\times n$ identity matrix. Since $S$ is
$k-$dimensional, $DH(z,0)|_{S}$ has $k$ multipliers equal to $1$ whose
corresponding eigenvectors span the tangent space $T_{z}S$. The remaining
$n-k$ multipliers $\mu_{j}(z),\ j=1,\ldots,n-k,$ may or may not lie on the
unit circle. In the following we refer to these $n-k$ multipliers as the non-
trivial multipliers of $DH(z,0)|_{S}$.

An important notion in the theory of fast-slow ODEs is the that of normal
hyperbolicity, which refers to the situation in which the non-trivial
eigenvalues of the linearized layer problem are all bounded away from the
imaginary axis. It is straightforward to define an analogous notion of normal
hyperbolicity for fast-slow maps (10) in terms of the non-trivial multipliers
$\mu_{j}(z)$.

######  Definition 2.6.

(Normal hyperbolicity) A point $z\in S$ is normally hyperbolic if the non-
trivial multipliers $\mu_{j}(z)$ do not lie on the unit circle, i.e

$|\mu_{j}(z)|\neq 1,\qquad j=1,\ldots,n-k.$

Additionally, a normally hyperbolic point $p$ is called:

  1. 1.

Attracting if $|\mu_{j}(z)|<1$ for all $j=1,\ldots,n-k$;

  2. 2.

Repelling if $|\mu_{j}(z)|>1$ for all $j=1,\ldots,n-k$;

  3. 3.

Saddle-type if $|\mu_{j}(z)|<1$ for $n_{a}<n-k$ multipliers $\mu_{j}$, and
$|\mu_{j}(z)|>1$ for $n_{r}<n-k$ multipliers, where $n_{a}+n_{r}=n-k$ and
$n_{a},n_{r}\geq 1$.

These definitions are extended to sets, i.e. a subset $S_{n}\subseteq S$ is
called normally hyperbolic if every point in $S_{n}$ is normally hyperbolic,
and a normally hyperbolic set $S_{n}$ is called attracting, repelling or
saddle-type if every point in $S_{n}$ is attracting, repelling or saddle-type
respectively.

(a) Attracting, $0<\mu<1$.

(b) Attracting, $-1<\mu<0$.

(c) Repelling, $\mu>1$.

(d) Repelling, $\mu<-1$.

(e) Saddle-type, $0<|\mu_{a}|<1$, $|\mu_{r}|>1$.

(f) Saddle-type, $0<|\mu_{a}|<1$, $|\mu_{r}|>1$.

Figure 1: Examples of normally hyperbolic critical manifolds $S$ in dimension
$n=3$. In (a)-(d) $\dim S=2$ and there exists a single non-trivial multiplier
$|\mu|\neq 1$. There are four generic possibilities depending on $\mu$: (a)
attracting and orientation preserving; (b) attracting and orientation
reversing; (c) repelling and orientation preserving; (d) repelling and
orientation reversing. In (e)-(f) $S$ is saddle-type with $\dim S=1$ and two
non-trivial multipliers $|\mu_{a}|<1$ and $|\mu_{r}|>1$. Two of four generic
possibilities are shown: (e) orientation preserving along stable/unstable
fibers $w^{s/u}_{loc}(z)$; (f) orientation reversing along $w^{s}_{loc}(z)$.

Figure 1 shows a number of representative scenarios in which the critical
manifold $S$ is normally hyperbolic and attracting, repelling or saddle-type.
Note, that the notion of normal hyperbolicity provided above applies equally
for orientation-preserving and orientation-reversing maps.

######  Remark 2.7.

Although we do not consider it further in this work, it is worthy to note that
there are three generic codimension-1 possibilities for a loss of normal
hyperbolicity along $S$ under additional parameter variation:

  1. 1.

Fold-type singularities: Crossing of a real non-trivial multiplier over $1$.

  2. 2.

Flip-type singularities: Crossing of a real non-trivial multiplier over $-1$.

  3. 3.

Neimark-Sacker-type singularities: Crossing of a pair of complex conjugate
non-trivial multipliers through $S^{1}\setminus\\{\pm 1\\}$.

With regards to the loss of normal hyperbolicity in fast-slow maps, the
analogy to the theory for fast-slow ODEs is not a direct one. For example, the
loss of normal hyperbolicity at a flip-type singularity has no direct analogue
in the corresponding ODE theory.

The following result provides a useful method for determining the non-trivial
multipliers $\mu_{j}$ in practice, and does not depend on a special choice of
coordinates.

######  Proposition 2.8.

We have the following equivalences:

  1. 1.

The subset of non-trivial eigenvalues of the matrix $NDf|_{S}$ coincides with
the set of eigenvalues of the $(n-k)\times(n-k)$ matrix $DfN|_{S}$.

  2. 2.

The set of non-trivial multipliers $\mu_{j}$ of the Jacobian matrix (12)
coincides with the set of eigenvalues of the $(n-k)\times(n-k)$ matrix
$I_{n-k}+DfN|_{S}$.

###### Proof.

The proof proceeds by analogy with the corresponding statement for ODEs in
[72, Lemma 3.3], with minor adaptations.

We work in a neighbourhood about a point $p\in S$. Assume without loss of
generality that $z=(x,y)^{\textnormal{T}}$ where $x\in\mathbb{R}^{k}$ and
$y\in\mathbb{R}^{n-k}$ are chosen such that locally, $D_{y}f$ is an
$(n-k)\times(n-k)$ regular (invertible) matrix; this is possible since
$Df|_{S}$ is regular by Assumption 1. In $(x,y)$ coordinates, the map (10) is
given by

$\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto\begin{pmatrix}\bar{x}\\\
\bar{y}\end{pmatrix}=\begin{pmatrix}x\\\
y\end{pmatrix}+\begin{pmatrix}N^{x}(x,y)\\\
N^{y}(x,y)\end{pmatrix}f(x,y)+\varepsilon\begin{pmatrix}G^{x}(x,y,\varepsilon)\\\
G^{y}(x,y,\varepsilon)\end{pmatrix},$ (13)

where $N^{x}(x,y)$, $N^{y}(x,y)$ are matrices of dimensions $k\times(n-k)$
respectively $(n-k)\times(n-k)$, and $G^{x}(x,y,\varepsilon)$,
$G^{y}(x,y,\varepsilon)$ are column vectors of length $k$ respectively $n-k$.

Applying the coordinate transformation

$v=f(x,y)$ (14)

with local (smooth) inverse $y=K(x,v)$ guaranteed to exist by the implicit
function theorem, locally rectifies the critical manifold $S$. Explicitly, we
obtain the following after local expansion about $v=0$, $\varepsilon=0$:

$\begin{split}v\mapsto\bar{v}&=f(\bar{x},\bar{y})\\\
&=f\left(x+N^{x}(x,y)v+\varepsilon
G^{x}(x,y,\varepsilon),y+N^{y}(x,y)v+\varepsilon
G^{y}(x,y,\varepsilon)\right)\\\ &=f\left(x,y\right)+DfN(x,y)v+\varepsilon
DfG(x,y,\varepsilon)+O(v^{2},v\varepsilon,\varepsilon^{2})\\\
&=v+DfN(x,K(x,v))v+\varepsilon
DfG(x,K(x,v),\varepsilon)+O(v^{2},v\varepsilon,\varepsilon^{2}),\end{split}$

and so the map becomes

$\begin{split}x&\mapsto\bar{x}=x+\tilde{N}^{x}(x,v)v+\varepsilon\tilde{G}^{x}(x,v,\varepsilon),\\\
v&\mapsto\bar{v}=v+DfN(x,K(x,v))v+\varepsilon
DfG(x,K(x,v),\varepsilon)+O(v^{2},v\varepsilon,\varepsilon^{2}),\end{split}$
(15)

where we denote $\tilde{N}^{x}(x,v):=N^{x}(x,K(x,v))$ and
$\tilde{G}^{x}(x,v,\varepsilon):=G(x,K(x,v),\varepsilon)$. Notice that (15)
has critical manifold $S=\\{v=0\\}$. The Jacobian evaluated along $S$ when
$\varepsilon=0$ is

$\begin{pmatrix}I_{k}&\tilde{N}^{x}(x,0)\\\
O_{n-k,k}&I_{n-k}+DfN|_{S}\end{pmatrix},$

which is block-diagonal with $k$ (trivial) multipliers equal to $1$, and $n-k$
non-trivial multipliers $\mu_{j}$ given by the eigenvalues of
$I_{n-k}+DfN|_{S}$.

Since both the coordinate transformation (14) and its inverse (which is
obtained via the implicit function theorem) are $C^{r}-$smooth with $r\geq 1$,
the eigenvalues (multipliers) along $S=\\{v=f(x,y)=0\\}$ are invariant under
the composed transformation, i.e. they are the same for map (10). The result
follows since the point $p\in S$ about which we applied our arguments was
arbitrary. ∎

By Proposition 2.8, the problem of calculating the $n-k$ non-trivial
multipliers of the $n\times n$ matrix $DH(z,0)|_{S}$ reduces to the problem of
calculating the multipliers of the $(n-k)\times(n-k)$ matrix
$I_{n-k}+DfN|_{S}$. Of course, this further reduces to the problem of
calculating the eigenvalues of $DfN|_{S}$.

######  Remark 2.9.

In the proof of Proposition 2.8 we utilised a special choice of coordinates,
in which the critical manifold $S$ is locally rectified along the $x-$axes.
However, Proposition 2.8 itself applies independently of the choice of
coordinates. The map (15) will often appear in proofs in later sections, and
we shall frequently make use of the existence of special coordinates in
proofs, but efforts will be made to ensure that ‘final results’ do not depend
on a choice of coordinates.

######  Remark 2.10.

Proposition 2.8 applies whether or not $S$ is normally hyperbolic. Hence
assertions (1)-(2) can be also used for identifying and classifying a loss of
normal hyperbolicity.

As an immediate consequence of Proposition 2.8, we have that for each non-
trivial multiplier $\mu_{j}$, there exists an eigenvalue $\lambda_{j}$ of the
matrix $DfN|_{S}$ such that

$\mu_{j}(z)=1+\lambda_{j}(z),\qquad z\in S.$ (16)

This leads to an alternative characterisation of normal hyperbolicity in terms
of the eigenvalues $\lambda_{j}$.

######  Corollary 2.11.

A point $z\in S$ is normally hyperbolic if and only if eigenvalues
$\lambda_{j}$ of the matrix $DFN|_{S}$ satisfy

$2\textup{Re}\,\lambda_{j}(z)\neq-|\lambda_{j}(z)|^{2},\qquad j=1,\ldots,n-k.$

Moreover, a normally hyperbolic point $z$ is attracting if

$2\textup{Re}\,\lambda_{j}(z)<-|\lambda_{j}(z)|^{2}$

for all $j=1,\ldots,n-k$, repelling if

$2\textup{Re}\,\lambda_{j}(z)>-|\lambda_{j}(z)|^{2}$

for all $j=1,\ldots,n-k$, and saddle-type otherwise.

###### Proof.

This follows directly from Proposition 2.8 and equation (16). ∎

Finally, we note that on the linear level, a normally hyperbolic critical
manifold $S_{n}\subseteq S$ induces a natural (pointwise) splitting

$T_{z}\mathbb{R}^{n}=T_{z}S_{n}\oplus\mathcal{N}_{z},\qquad z\in S_{n},$ (17)

with $\mathcal{N}_{z}:=\text{span}\\{N^{i}(z)\\}_{j=1,\ldots,n-k}=E^{s}(z)\cup
E^{u}(z)$, where $N^{i}(z)$ denotes the $i$’th column of $N(z)$ and
$E^{s/u}(z)$ denote stable/unstable eigenspaces at $z$ respectively. The
pointwise splitting (17) leads to the global splitting

$\mathbb{R}^{n}\cong T\mathbb{R}^{n}=TS_{n}\oplus\mathcal{N},$ (18)

where $TS_{n}=\cup_{z\in S_{n}}T_{z}S$ and $\mathcal{N}=\cup_{z\in
S_{n}}\mathcal{N}_{z}$ denote the tangent bundle and transverse linear fiber
bundle associated to $S_{n}$, respectively.

Figure 2: Stable/unstable manifolds $W^{s/u}_{loc}(S_{n})$ associated to a
normally hyperbolic critical manifold $S_{n}$ are foliated by lower
dimensional stable/unstable manifolds $w^{s/u}_{loc}(z)$ associated to base
points $z\in S_{n}$. The case of a $1-$dimensional saddle-type critical
manifold in dimension $n=3$ is shown here ($S_{n}$ is shown in green). In this
case the manifolds $W^{s/u}_{loc}(S_{n})$ (shown in blue/red) are
$2-$dimensional and foliated by the $1-$dimensional stable/unstable manifolds
$w^{s/u}_{loc}(z)$ associated to base points $z\in S_{n}$. Representative
stable/unstable fibers $w^{s/u}(z_{i})$ with base points $z_{i}\in S_{n}$ for
$i=1,2$ are also shown.

On the nonlinear level, a normally hyperbolic critical manifold
$S_{n}\subseteq S$ induces a local foliation of the adjacent space by its
local stable and unstable manifolds $w_{loc}^{s}(z)$ and $w_{loc}^{u}(z)$. We
denote induced stable and unstable foliations by

$W_{loc}^{s}(S_{n}):=\bigcup_{z\in S_{n}}w_{loc}^{s}(z),\qquad
W_{loc}^{u}(S_{n}):=\bigcup_{z\in S_{n}}w_{loc}^{u}(z),$ (19)

respectively. Note that if $|\mu_{j}|<1$ for $n_{a}$ non-trivial multipliers
and $|\mu_{j}|>1$ for $n_{r}$ non-trivial multipliers ($n_{a}+n_{r}=n-k$),
then $W_{loc}^{s}(S_{n})$ is $(k+n_{a})-$dimensional and $W_{loc}^{u}(S_{n})$
is $(k+n_{r})-$dimensional. By definition, the fibers $w_{loc}^{s/u}(z)$ and
hence the induced foliations in (19) are locally invariant under the layer map
(8). Together, $W^{s/u}_{loc}(S_{n})$ form a local nonlinear fiber bundle,
whose linear part coincides with the transverse linear fiber bundle
$\mathcal{N}$. The case of a saddle-type normally hyperbolic critical manifold
in $\mathbb{R}^{3}$ with $n_{s}=n_{u}=1$ is sketched in Figure 2.

######  Remark 2.12.

In the case that $S$ is normally hyperbolic, the transverse linear fiber
bundle $\mathcal{N}$ is related to the normal bundle associated to the tangent
bundle $TS$ by the smooth coordinate transformation described in Remark 2.23
below. Thus it is common to find the terms “transverse bundle” and “normal
bundle” used interchangeably in the literature. The former terminology is
preferred in this work in order to emphasise the coordinate-independence of
the formalism.

###  2.2 The reduced and $m$’th iterate maps

In the context of fast-slow ODEs one obtains a second, non-equivalent limiting
problem – the reduced problem – by considering the singular limit
$\varepsilon\to 0$ taken with respect to the system of fast-slow ODEs posed on
the so-called slow time $\tau=\varepsilon t$. This reduced problem induces a
flow on normally hyperbolic submanifolds of $S$ on the slow timescale $\tau$.
By analogy, one might expect a similar equivalence between maps of the form
(10) and ‘slow’ maps of the form

$z\mapsto z+\frac{1}{\varepsilon}N(z)f(z)+G(z,\varepsilon).$

Unfortunately, such an equivalence is not available for maps, since there is
in general no analogue of the time rescaling which would render the maps

$\bar{z}-z\qquad\text{and}\qquad\varepsilon^{-1}(\bar{z}-z)$

equivalent for each $0<\varepsilon\ll 1$. It is natural to ask, then, whether
it is still possible to derive a ‘reduced map’ which describes the limiting
dynamics on (normally hyperbolic submanifolds of) the critical manifold $S$,
thereby providing information on the limiting dynamics which is not present in
the layer map (8).

In this section we show that such a map can be derived, though it must be
formulated as a map governing the leading order dynamics on a locally
invariant $C^{r}-$smooth slow manifold $S_{\varepsilon}$ which converges to
$S$ as $\varepsilon\to 0$. Under certain nondegeneracy conditions, sequential
iterates of this map on the slow manifold $S_{\varepsilon}$ are
$O(\varepsilon)-$close to each other. We also derive the reduced $m$’th
iterate map on $S_{\varepsilon}$ induced by repeated iteration of (10).
Interestingly, this map can be related to a discretization of the ODE reduced
problem associated to fast-slow ODEs in [72] if the number of iterates $m$ is
comparable to $\varepsilon^{-1}$.

######  Remark 2.13.

As stated above, in the following we need to assume the existence of an
invariant slow manifold in order to derive both the reduced and $m$’th iterate
maps. The existence of slow manifolds under suitable (normally hyperbolic)
conditions will be treated in Section 3 (see Theorems 3.1 and 3.2), and does
not depend on the existence of a reduced map.

####  2.2.1 The reduced map

Consider again a connected, normally hyperbolic submanifold $S_{n}\subseteq
S$. Due to the splitting (18), there exists a unique projection operator

$\Pi^{S_{n}}_{\mathcal{N}}:T\mathbb{R}^{n}=TS_{n}\oplus\mathcal{N}\to TS_{n},$
(20)

which projects vectors in $T\mathbb{R}^{n}$ onto their component in $TS_{n}$
along the direction of $\mathcal{N}$, see Figure 3. This allows one to isolate
components of the map contributing to dynamics in $TS_{n}$. The following
result should be compared with the characterisation of the reduced problem in
fast-slow ODEs presented in [72, Def. 3.8], see also [17, 20, 40, 49].

Figure 3: Action of the oblique projection operator
$\Pi^{S_{n}}_{\mathcal{N}}$ along $\mathcal{N}$ onto $TS_{n}$ in the case of a
$1-$dimensional attracting critical manifold $S_{n}$ (green). The splitting
(18) implies that the leading order perturbation $G(z,0)$ at each point $z\in
S_{n}$ can be decomposed into components in the tangent bundle $T_{z}S_{n}$,
and the (transverse) linear fast fiber $\mathcal{N}_{z}$ tangent to the
nonlinear fast fiber $w^{s}_{loc}(z)$ shown in blue in the figure. The
component in $T_{z}S_{n}$ is given by $\Pi_{\mathcal{N}}^{S_{n}}G(z,0)\in
T_{z}S_{n}$, and sufficient to determine the reduced map (24), see also
Proposition 2.14.

######  Proposition 2.14.

Fix $\varepsilon\in[0,\varepsilon_{0})$ and denote by $S_{n,\varepsilon}$ a
locally invariant slow manifold perturbing from $S_{n}$, as described by
Theorem 3.1. Then for $\varepsilon_{0}$ sufficiently small we have

$\bar{z}|_{S_{n,\varepsilon}}=z+\varepsilon\Pi^{S_{n}}_{\mathcal{N}}G(z,0)+O(\varepsilon^{2}),\qquad
z\in S_{n}.$ (21)

Moreover, the unique projection operator $\Pi^{S_{n}}_{\mathcal{N}}$ has
matrix representation

$\Pi^{S_{n}}_{\mathcal{N}}=I_{n}-N(DfN)^{-1}Df|_{S_{n}}.$ (22)

###### Proof.

Without loss of generality, we make the same preliminary assumptions as in the
proof of Proposition 2.8 and consider again the map (13) in a neighbourhood
about some $p\in S_{n}\subseteq S$. In particular, we assume that the matrix
$D_{y}f$ is locally regular so that the critical manifold $S$ has a graph
representation $y=\varphi_{0}(x)$.

To obtain the map (21), we first obtain a local graph form for the invariant
slow manifold by substituting the ansatz

$S_{n,\varepsilon}:y=\varphi_{\varepsilon}(x)=\varphi_{0}(x)+\varphi_{1}(x)\varepsilon+O(\varepsilon^{2}),$

where $f(x,\varphi_{0}(x))=0$, into the invariance equation

$\bar{y}=\varphi_{\varepsilon}(\bar{x})=\varphi_{\varepsilon}(x)+N^{y}(x,\varphi_{\varepsilon}(x))f(x,\varphi_{\varepsilon}(x))+\varepsilon
G^{y}(x,\varphi_{\varepsilon}(x),\varepsilon).$

By redefining $N(x,y)$ if necessary, we may assume that

$f(x,y)=y-\varphi_{0}(x),$

without loss of generality. By expanding in $\varepsilon$ and requiring that
$(x,y)\in S_{\varepsilon}\implies(\bar{x},\bar{y})\in S_{\varepsilon}$, we
obtain the following invariance equation after matching terms at
$O(\varepsilon)$:

$\begin{split}\varphi_{1}(x)&+N^{y}(x,\varphi_{0}(x))\varphi_{1}(x)+G^{y}(x,\varphi_{0}(x),0)\\\
&=\varphi_{1}(x)+D_{x}\varphi_{0}(x)N^{x}(x,\varphi_{0}(x))\varphi_{1}(x)+D_{x}\varphi_{0}(x)G^{x}(x,\varphi_{0}(x),0).\end{split}$

Rearranging for $\varphi_{1}(x)$ and using

$\begin{split}DfN|_{S}&=-D_{x}\varphi_{0}(x)N^{x}(x,\varphi_{0}(x))+N^{y}(x,\varphi_{0}(x)),\\\
DfG|_{S}&=-D_{x}\varphi_{0}(x)G^{x}(x,\varphi_{0}(x),0)+G^{y}(x,\varphi_{0}(x),0),\end{split}$

yields

$S_{\varepsilon}:y=\varphi_{\varepsilon}(x)=\varphi_{0}(x)-\varepsilon(DfN)^{-1}DfG|_{S}+O(\varepsilon^{2}).$
(23)

Substituting (23) into (13) and expanding again in $\varepsilon$ yields (21)
with $\Pi^{S_{n}}_{\mathcal{N}}$ given by the matrix (22). ∎

######  Remark 2.15.

The expression (23) obtained in the proof above is of interest its own right,
as it provides a general formula for the slow manifold parameterization up to
$O(\varepsilon)$ in terms of $N$, $f$ and $G$ in the case that the critical
manifold $S$ is given as a graph $y=\varphi_{0}(x)$.

Proposition 2.14 allows for the following definition.

######  Definition 2.16.

(Reduced map) The map

$z\mapsto\bar{z}=z+\varepsilon\Pi^{S_{n}}_{\mathcal{N}}G(z,0),\qquad z\in
S_{n},$ (24)

obtained by truncating (24) at $O(\varepsilon^{2})$ is called the reduced map.

Similarly to the theory for fast-slow ODEs, the reduced map describes the
leading order dynamics on locally invariant slow manifolds perturbing from
normally hyperbolic submanifolds of $S$. In contrast to the ODE theory, an
$\varepsilon-$independent expression for the right-hand-side is not available,
since there is in general no way to ‘divide out’ the factor of $\varepsilon$
while preserving topological conjugacy. Fortunately, this is no hindrance in
applications, since what is really needed is a calculable asymptotic
approximation for the flow along locally invariant slow manifolds with
$0<\varepsilon\ll 1$. This is provided by the formulation of the reduced map
(24).

######  Remark 2.17.

A similar formulation of the reduced problem also exists in the ODE setting,
i.e. there too, the reduced vector field is precisely the leading order vector
field for the dynamics on Fenichel slow manifolds, occurring at
$O(\varepsilon)$ on the fast timescale.

######  Remark 2.18.

In the special case of maps induced via discretization of a fast-slow ODE, a
reduced map having an $\varepsilon-$independent right-hand-side can typically
be obtained by a rescaling of the discretization/step parameter $h>0$. For
example, Euler-discretization of a general fast-slow ODE

$z^{\prime}=N(z)f(z)+\varepsilon G(z,\varepsilon),\qquad
z\in\mathbb{R}^{n},\qquad 0<\varepsilon\ll 1,$

leads to the map

$\frac{\bar{z}-z}{h}=N(z)f(z)+\varepsilon G(z,\varepsilon).$

Rescaling $h=\varepsilon^{-1}\tilde{h}$ leads to the reduced map

$\frac{\bar{z}-z}{\tilde{h}}=\Pi^{S_{n}}_{\mathcal{N}}G(z,0),\qquad z\in
S_{n},$

which has no $\varepsilon-$dependence in the right-hand-side. Euler
discretizations of this kind are considered further in Section 5.2.

Finally we note that similarly to the ODE case, the projection operator
$\Pi^{S_{n}}_{\mathcal{N}}$ is not defined at fold-type singularities since
the quantity $\Pi^{S_{n}}_{\mathcal{N}}G(z,0)$ blows up. It is however defined
at flip and Neimark-Sacker-type singularities. We do not consider these issues
(which relate to the loss of normal hyperbolicity) in detail in this work. See
however Section 6 for further discussion.

####  2.2.2 The $m$’th iterate map

In a neighbourhood of the critical manifold $S$ it is also possible to derive
the form of the $m$’th iterate map induced by repeated iteration of the map
(10) using local asymptotic self-similarity properties of the map. As before,
$S_{n}$ denotes a compact normally hyperbolic submanifold of $S$.

######  Proposition 2.19.

For all $\varepsilon\in[0,\varepsilon_{0})$ with $\varepsilon_{0}$
sufficiently small, the $m$’th iterate map induced by (10) in a tubular
neighbourhood of $S$ takes the form of a non-autonomous map

$z\mapsto\bar{z}^{m}=z+m\left(N(z)f(z)+\varepsilon
G(z,\varepsilon)\right)+R(z,\varepsilon,m),$ (25)

where for all fixed $m\in\mathbb{N}_{+}$ the higher order term
$R(z,\varepsilon,m)$ satisfies

  1. (i)

$R(z,\varepsilon,m)=O(\varepsilon^{2},\varepsilon f(z),f(z)^{2})$;

  2. (ii)

$R(z,\varepsilon,1)\equiv O_{n}$;

  3. (iii)

$R(z,0,m)|_{S_{n}}\equiv O_{n}$;

  4. (iv)

$R(z,\varepsilon,m)\to O_{n}$ as $(\varepsilon,m)\to(0,\infty)$.

In particular, the reduced $m$’th iterate map on $S_{n,\varepsilon}$ is given
by

$z\mapsto\bar{z}^{m}=z+\varepsilon m\Pi^{S_{n}}_{\mathcal{N}}G(z,0),\qquad
z\in S_{n},$ (26)

where $\Pi^{S_{n}}_{\mathcal{N}}$ is the projection operator (22).

###### Proof.

We work in a tubular neighbourhood of $S$ within which Taylor expansion about
$f(z)=O_{n-k}$ is valid. Repeated iteration of the map (10) leads to the
following asymptotically self-similar sequence:

$\begin{split}\bar{z}&=z+N(z)f(z)+\varepsilon G(z,\varepsilon),\\\
\bar{z}^{2}&=z+2\left(N(z)f(z)+\varepsilon
G(z,\varepsilon)\right)+R(z,\varepsilon,2),\\\ &\ \vdots\\\
\bar{z}^{m}&=z+m\left(N(z)f(z)+\varepsilon
G(z,\varepsilon)\right)+R(z,\varepsilon,m),\end{split}$

where $m\in\mathbb{N}_{+}$. For fixed $m$, the properties (i)-(iii) of the
remainder term $R(z,\varepsilon,m)$ are shown by induction on $m$, and omitted
here for brevity. In order to prove property (iv), we adopt a sequential,
componentwise notation

$R(z,\varepsilon,m):=R_{m}(z,\varepsilon)=(R_{m}^{1}(z,\varepsilon),\ldots,R_{m}^{n}(z,\varepsilon))^{\textnormal{T}}\in\mathbb{R}^{n},\qquad
m\in\mathbb{N}_{+},$

and consider the real-valued sequences $(R^{i}_{m})_{m\in\mathbb{N}_{+}}$
defined by the component functions
$R_{m}^{i}:\mathbb{R}^{n}\times[0,\varepsilon_{0})\to\mathbb{R}$,
$i=1,\ldots,n$. Since by property (iii) (which holds for all fixed
$m\in\mathbb{N}_{+}$) we have that

$\lim_{m\to\infty}R^{i}_{m}(z,0)=0,\qquad z\in S_{n},$

uniformly in $(z,\varepsilon)$, and by continuity of $R$ we have that

$\lim_{\varepsilon\to 0}R^{i}_{m}(z,\varepsilon)=R^{i}_{m}(z,0)=0,\qquad z\in
S_{n},$

pointwise in $(z,\varepsilon)$, it follows by an application of the Moore-
Osgood theorem [56] that the double limit is defined and commutes such that

$\lim_{\varepsilon\to
0}\lim_{m\to\infty}R^{i}_{m}(z,\varepsilon)=\lim_{m\to\infty}\lim_{\varepsilon\to
0}R^{i}_{m}(z,\varepsilon)=0,\qquad z\in S_{n}.$

Since the above holds for each $i=1,\ldots,n$, it follows that
$R(z,\varepsilon,m)|_{S_{n}}\to 0$ as $(\varepsilon,m)\to(0,\infty)$, as
required.

It remains to show that the reduced $m$’th iterate map on $S_{n,\varepsilon}$
is given by (26). Since the expression (26) is pointwise, it suffices to show
it in the local $z=(x,y)^{\textnormal{T}}$ coordinates of the map (13) used in
proof of Propositions 2.8 and 2.14. Specifically, the same preliminary
assumptions used in the proof of Propositions 2.8 and 2.14 lead to the
following local formulation of the $m$’th iterate map (25) near $S$:

$\begin{split}x&\mapsto\bar{x}^{m}=x+m\left(N^{x}(x,y)(y-\varphi_{0}(x))+\varepsilon
G^{x}(x,y,\varepsilon)\right)+R^{x}(x,y,\varepsilon,m),\\\
y&\mapsto\bar{y}^{m}=y+m\left(N^{y}(x,y)(y-\varphi_{0}(x))+\varepsilon
G^{y}(x,y,\varepsilon)\right)+R^{y}(x,y,\varepsilon,m),\end{split}$

where we assume that $f(x,y)=y-\varphi_{0}(x)$ as in the proof of Proposition
2.14 (recall that this can be achieved by redefining $N$ if necessary), and
where the remainder terms satisfy

$\begin{split}R^{x}(x,y,\varepsilon,m)&=O\left(\varepsilon^{2},\varepsilon(y-\varphi_{0}(x)),(y-\varphi_{0}(x))^{2}\right),\\\
R^{y}(x,y,\varepsilon,m)&=O\left(\varepsilon^{2},\varepsilon(y-\varphi_{0}(x)),(y-\varphi_{0}(x))^{2}\right),\end{split}$
(27)

for each fixed $m\in\mathbb{N}_{+}$ due to the remainder property (i). By
invariance, we can simply restrict the map to $S_{n,\varepsilon}$, which in
the chosen coordinates has graph representation (23). We obtain

$\begin{split}\begin{pmatrix}\bar{x}^{m}\\\
\bar{y}^{m}\end{pmatrix}&=\begin{pmatrix}x\\\ y\end{pmatrix}+\varepsilon
m\begin{pmatrix}-N^{x}(DfN)^{-1}DfG+G^{x}\\\
-N^{y}(DfN)^{-1}DfG+G^{y}\end{pmatrix}\bigg{|}_{y=\varphi_{0}(x),\varepsilon=0}+O(\varepsilon^{2})\\\
&=\begin{pmatrix}x\\\ y\end{pmatrix}+\varepsilon
m\left(I_{n}-N(DfN)^{-1}Df\right)G\big{|}_{y=\varphi_{0}(x),\varepsilon=0}+O(\varepsilon^{2})\\\
&=\begin{pmatrix}x\\\ \varphi_{0}(x)\end{pmatrix}+\varepsilon
m\Pi_{\mathcal{N}}^{S_{n}}G(x,\varphi_{0}(x),0)+O(\varepsilon^{2}),\end{split}$
(28)

after expanding in $\varepsilon$, using the fact that
$R^{x}(x,\varphi_{\varepsilon}(x),\varepsilon,m),R^{y}(x,\varphi_{\varepsilon}(x),\varepsilon,m)=O(\varepsilon^{2})$
for each fixed $m\in\mathbb{N}_{+}$ by equation (27) above, and substituting
(22) for $\Pi_{\mathcal{N}}^{S_{n}}$ in the final equality. The expression in
(26) follows from (28) after truncating the $O(\varepsilon^{2})$ terms in line
with Definition 2.16. ∎

(a) Reduced map on $S_{n,\varepsilon}$.

(b) $m$’th iterate map on $S_{n,\varepsilon}$.

Figure 4: Reduced and $m$’th iterate maps on $S_{n,\varepsilon}$ defined in
equations (24) and (26) respectively. The normally hyperbolic critical
manifold and the corresponding perturbed slow manifold are shown again in
green and shaded green respectively. The dynamics of the map (10) within
$S_{n,\varepsilon}$ are governed to leading order in $\varepsilon$ by the
reduced map (24). Iterates are generically separated by distances of
$O(\varepsilon)$, as shown in (a). Iterates of the $m$’th iterate reduced map
(26) shown in (b) can be separated by $O(1)$ distances if the number of
iterates $m$ is comparable to $\varepsilon^{-1}$. For both reduced and $m$’th
iterate maps, non-trivial dynamics is only possible for $0<\varepsilon\ll 1$.
See however Remark 2.20, which describes a possible approach to identifying
non-trivial dynamics on $S$ for $\varepsilon=0$ in the dual limit
$(\varepsilon,m)\to(0,\infty)$.

See Figure 4 for a comparison of reduced and $m$’th iterate maps. If the
number of iterates $m\in\mathbb{N}_{+}$ is comparable to $\varepsilon^{-1}$,
then iterates of the $m$’th iterate map are generically separated by
$O(m\varepsilon)=O(1)-$distances. In essence, the $\varepsilon-$dependence
cancels if one ‘speeds up’ the map by considering only every $m$’th iterate
with $m=\varepsilon^{-1}\tilde{m}$ for some $\tilde{m}>0$. This has a similar
effect to the time rescaling $t=\varepsilon^{-1}\tau$ in fast-slow ODEs. In
fact the $m$’th iterate map (26) relates directly to the ODE reduced problem
$z^{\prime}=\Pi_{\mathcal{N}}^{S_{n}}G(z,0)$ associated to general fast-slow
systems (1) (see [72, Def. 3.8]), via the Euler discretization of the latter.
Specifically, the map (26) can be rewritten as

$\frac{{\bar{z}}^{m}-z}{\tilde{m}}=\Pi_{\mathcal{N}}^{S_{n}}G(z,0),$ (29)

which coincides with an Euler discretization of the ODE reduced problem if the
step size $\tilde{m}>0$ satisfying
$\varepsilon^{-1}\tilde{m}=m\in\mathbb{N}_{+}$; recall also Remark 2.18.

######  Remark 2.20.

The preceding discussion pertains to a relationship between the Euler
discretization (29) and the $m$’th iterate reduced map (26), which is obtained
by truncating the map (28) at $O(\varepsilon^{2})$. In order to rigorously
prove a relationship between the discretized (continuous-time) reduced problem
and an $m$’th iterate map on $S$ for $\varepsilon=0$, one must define a
suitable embedding of the map (28) with $m\in\mathbb{N}_{+}$ into the
parameter-dependent family

$\bar{z}_{\mathbb{R}}=z+\varepsilon\zeta\Pi_{\mathcal{N}}^{S_{n}}G(z,0)+R(z,\varepsilon,\zeta),\qquad
z\in S_{n},\qquad\zeta>0,$

and study the limit $\varepsilon\to 0$ in the case that
$\zeta=\varepsilon^{-1}\tilde{\zeta}$ with fixed $\tilde{\zeta}>0$ (this
corresponds to a dual limit $(\varepsilon,m)\to(0,\infty)$ in (28)). This is
left for future work.

###  2.3 Fast-slow maps in standard form

In this section we consider a particularly important subclass of fast-slow
maps (10), namely, fast-slow maps in the so-called standard form

$\begin{split}x&\mapsto\bar{x}=x+\varepsilon\tilde{g}(x,y,\varepsilon),\\\
y&\mapsto\bar{y}=y+\tilde{f}(x,y,\varepsilon),\end{split}$ (30)

where the ‘fast-slow’ structure is explicit in the $\varepsilon-$factorisation
of $\bar{x}-x=\varepsilon\tilde{g}(x,y,\varepsilon)$, which leads to a global
separation into ‘slow variables’ $x\in\mathbb{R}^{k}$, and ‘fast variables’
$y\in\mathbb{R}^{n-k}$. Of course, timescale terminology like ‘fast-slow’
should be understood here only by analogy to the corresponding ODE systems:
$x$ is a ‘slow variable’ in the sense that successive iterates will
generically be $O(\varepsilon)-$close to one another, while $y$ is a ‘fast
variable’ in the sense that successive iterates will generically be separated
by distances of $O(1)$. Many applications arise naturally in the standard form
(30), including those considered in detail in Sections 5.1 and 5.3. In
particular, allowing for slow evolution in a systems parameters leads to a
fast-slow map in standard form (30).

In the following we assume that
$\tilde{g}:\mathbb{R}^{n}\times[0,\varepsilon_{0})\to\mathbb{R}^{k}$ and
$\tilde{f}:\mathbb{R}^{n}\times[0,\varepsilon_{0})\to\mathbb{R}^{n-k}$ in (30)
are $C^{r\geq 1}-$smooth in all arguments (although $\tilde{g}$ need only be
$C^{0}$ in $\varepsilon$), in which case the map (30) can be written in the
more general form (10) by expanding

$\tilde{f}(x,y,\varepsilon)=\tilde{f}(x,y,0)+\varepsilon\tilde{f}_{rem}(x,y,\varepsilon)$

so that

$z=\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto\begin{pmatrix}\bar{x}\\\
\bar{y}\end{pmatrix}=\begin{pmatrix}x\\\
y\end{pmatrix}+\begin{pmatrix}O_{k,n-k}\\\
I_{n-k}\end{pmatrix}\tilde{f}(x,y,0)+\varepsilon\begin{pmatrix}\tilde{g}(x,y,\varepsilon)\\\
\tilde{f}_{rem}(x,y,\varepsilon)\end{pmatrix}.$

Thus, by writing $z=(x,y)^{\textnormal{T}}\in\mathbb{R}^{n}$, the class of
maps (30) can be considered as an important subclass of fast-slow maps (10)
for which we have

$N(z)=\begin{pmatrix}O_{k,n-k}\\\ I_{n-k}\end{pmatrix},\qquad
f(z)=\tilde{f}(x,y,0),\qquad
G(z,\varepsilon)=\begin{pmatrix}\tilde{g}(x,y,\varepsilon)\\\
\tilde{f}_{rem}(x,y,\varepsilon)\end{pmatrix}.$ (31)

Assumptions 1-2 are satisfied for fast-slow maps in standard form (30) if the
level set $S=\\{(x,y):\tilde{f}(x,y,0)=O_{n-k}\\}$ exists. In the following we
provide corollaries of the more general notions derived so far for the special
case of standard form maps (30).

######  Remark 2.21.

The class of fast-slow maps in standard form (30) should be distinguished from
the class of maps in the form

$\begin{split}x&\mapsto\bar{x}=\varepsilon\tilde{g}(x,y,\varepsilon),\\\
y&\mapsto\bar{y}=\tilde{f}(x,y,\varepsilon),\end{split}$ (32)

which also arise in applications; see e.g. the Hénon map in [42, eqn. (14.11)]
or the detailed analysis of invariant manifolds and corresponding foliations
in 2-dimensional maps (32) in [66, Appendix A]. Generically, the maps (32) are
regularly (as opposed to singularly) perturbed in the sense of Definition 2.1,
since the critical set
$\\{(O_{k}^{\textnormal{T}},y):f(O_{k}^{\textnormal{T}},y,0)=O_{n-k}\\}$ is
generically empty or comprised entirely of isolated points
$(O_{k}^{\textnormal{T}},y_{\ast})$ for which $y_{\ast}$ satisfies
$\tilde{f}(O_{k}^{\textnormal{T}},y_{\ast},0)=O_{n-k}$. It follows that the
theory developed herein does not in general apply for maps in the form (32),
since they may not satisfy Assumption 1.

####  2.3.1 Layer map

The layer map for (30) is

$\begin{split}&x\mapsto\bar{x}=x,\\\
&y\mapsto\bar{y}=y+\tilde{f}(x,y,0).\end{split}$ (33)

Notice that the slow variables $x\in\mathbb{R}^{k}$ become parameters. This is
a direct consequence of the separation of slow and fast variables in (30), and
not true generally for fast-slow maps (10), recall the layer map (8).
Geometrically, we see that the subclass of fast-slow maps in standard form
(30) consists of the maps (10) for which the fast foliation defined by the
layer map (8) is globally rectified; see Figure 5.

(a) Standard.

(b) Non-standard only.

Figure 5: Fast-slow maps in the standard form (30) can be characterised
geometrically as the subclass of fast-slow maps in general form (10) for which
the fast foliation has been or can be globally rectified. Such a situation is
sketched in (a). In (b) we show an example of a fast-slow map with an isolated
fixed point $Q$. Such a map cannot be written in standard form (30) since the
fast foliation cannot be globally rectified.

The critical manifold is

$S=\left\\{(x,y):\tilde{f}(x,y,0)=f(x,y)=O_{n-k}\right\\},$

and it follows from Proposition 2.8 that the $n-k$ non-trivial multipliers
along $S$ coincide with the $n-k$ eigenvalues of the matrix

$I_{n-k}+NDf|_{S}=I_{n-k}+D_{y}f|_{S}.$

####  2.3.2 Reduced map

It follows from equations (22) and (31) that the projection operator
$\Pi^{S}_{\mathcal{N}}$ associated with the map (30) has matrix representation

$\Pi^{S}_{\mathcal{N}}=I_{n}-N(DfN)^{-1}Df=\begin{pmatrix}I_{k}&O_{k,n-k}\\\
-(D_{y}f)^{-1}D_{x}f&O_{n-k,n-k}\end{pmatrix}.$

Hence the reduced map is given by

$\begin{split}&x\mapsto\bar{x}=x+\varepsilon\tilde{g}(x,y,0),\\\
&y\mapsto\bar{y}=y-\varepsilon(D_{y}f)^{-1}D_{x}f\tilde{g}(x,y,0),\end{split}\qquad(x,y)\in
S.$

and for each fixed $0<\varepsilon\ll 1$ there is a reduced
$\varepsilon^{-1}\tilde{m}=m$’th iterate map (26) defined by

$\begin{split}&x\mapsto\bar{x}^{m}=x+\tilde{m}\tilde{g}(x,y,0),\\\
&y\mapsto\bar{y}^{m}=y-\tilde{m}(D_{y}f)^{-1}D_{x}f\tilde{g}(x,y,0),\end{split}\qquad(x,y)\in
S.$

One can clearly see a correspondence with the well-known expression for the
reduced problem for fast-slow ODEs in standard form, see e.g. [34, 36, 42].

####  2.3.3 Local transformation to standard form

Similarly to the ODE case, there is a local equivalence between fast-slow maps
in standard and non-standard form. We stress again that such an equivalence is
only local, and valid only in sufficiently small neighbourhoods about points
$p\in S$.

######  Proposition 2.22.

Consider the fast-slow map (10) and let $p\in S$ be a normally hyperbolic
point. For all $\varepsilon\in[0,\varepsilon_{0})$ with $\varepsilon_{0}$
sufficiently small, there exists a smooth invertible local coordinate
transformation such that (10) takes the standard form (30) in a sufficiently
small neighbourhood about $p$.

###### Proof.

We may proceed via arguments similar to those applied in the proof of
corresponding results for ODEs [72, Section 3.7]. We assume the same local
conditions as in the proof of Proposition 2.8, in particular that the
$(n-k)\times(n-k)$ matrix $D_{y}f|_{S}$ is regular so that $S$ is given
locally as a graph $y=\varphi_{0}(x)$. Due to normal hyperbolicity, the
nonlinear fast fibers $\mathcal{F}$ are given as constant level sets
$L(x,y)=c$, for a $C^{r}-$smooth function
$L:\mathbb{R}^{k}\times\mathbb{R}^{n-k}\to\mathbb{R}^{n-k}$ such that the
$(n-k)\times k$ matrix $D_{x}L$ is locally regular and the following
invariance property is satisfied:

$L(x+N^{x}(x,y)f(x,y),y+N^{y}(x,y)f(x,y))=L(x,y).$ (34)

The function $L(x,y)$ is guaranteed to exist by the center manifold theorem,
since for $\varepsilon=0$ each nonlinear fiber $\mathcal{F}$ is just the union
of stable and unstable manifolds for $p\in S$ considered as a fixed point of
the layer map (8). The foliation for $\varepsilon=0$ can be rectified via the
coordinate transformation

$s=L(x,y),$

which has a local inverse $x=K(s,y)$ by the inverse function theorem since
$D_{x}L$ is locally regular. Using the invariance property (34) and expanding
about $\varepsilon=0$ we obtain the map

$\begin{split}s&\mapsto s+\varepsilon
DLG(K(s,y),y,\varepsilon)+O(\varepsilon^{2}),\\\ y&\mapsto
y+N^{y}(K(s,y),y)f(K(s,y),y)+\varepsilon
G^{y}(K(s,y),y,\varepsilon),\end{split}$ (35)

where $DLG=D_{x}LG^{x}+D_{y}LG^{y}$. The map (35) is in the standard form
(30). ∎

######  Remark 2.23.

In the proof of Proposition 2.22 we rectified the fast foliation in a
neighbourhood of a normally hyperbolic point of $S$. By a subsequent
application of the coordinate transformation used in the proof of Proposition
2.8, one can also rectify the critical manifold $S$ in such neighbourhoods in
order to obtain a local normal form analogous to the well-known (local)
Fenichel normal form [17, 34, 42] in the continuous-time setting.

######  Remark 2.24.

Proposition 2.22 asserts the existence of a local coordinate transformation
putting general non-standard form maps (10) locally (not globally) into
standard form (30). It is worthy to note that the problem of obtaining an
explicit form of the function $L(x,y)$ used in the transformation for a given
application is typically very difficult or intractable. A direct application
of the coordinate-independent theory developed in earlier sections is
typically preferred on practical grounds in such cases.

##  3 Slow manifold theorems

In this section we state the main results. Results pertaining to persistence
of normally hyperbolic critical manifolds as locally invariant slow manifolds,
as well as persistence of the stable/unstable manifolds (19) and their locally
invariant foliations are given. For expository reasons we have decided to
state the results via a series of independent statements, similarly to the
presentation of Fenichel’s theorems for fast-slow ODEs in [34]. We shall also
– again similarly again to [34] – present two versions of the result
describing perturbations of the critical manifold, the latter being a
specialisation to the case in which the obtained slow manifold/foliation has a
graph representation. Though the latter (graph) formulation is less general
and easily derived from the former (manifold) formulation, it is worthwhile to
present both here for two reasons. Firstly, the critical manifold $S$ is
frequently a graph in applications. Secondly, a graph formulation is always
achievable locally. This fact is also leveraged in the proofs in Section 4,
where we shall typically restrict to the analysis of local graph formulations
since the more global statements for compact manifolds are obtained via
standard arguments based on a partition of unity.

We begin with the existence of locally invariant slow manifolds obtained as
perturbations of the critical manifold. We shall assume throughout that the
critical manifold $S$ is normally hyperbolic. This saves us from restricting
to normally hyperbolic submanifolds $S_{n}\subseteq S$ as in Section 2. All
main results are stated for general maps (10), stated again here for
convenience,

$z\mapsto\bar{z}=H(z,\varepsilon)=z+N(z)f(z)+\varepsilon G(z,\varepsilon),$
(36)

where $H(z,\varepsilon)$ is $C^{r\geq 1}-$smooth on an open set
$\mathcal{U}\times(-\varepsilon_{0},\varepsilon_{0})$ with
$\mathcal{U}\subset\mathbb{R}^{n}$ and subject to Assumptions 1-2.

######  Theorem 3.1.

(Existence of slow manifolds) Consider the map (36) under Assumptions 1-2. Let
$S$ be a compact, connected and normally hyperbolic critical manifold. Then
there is an $\varepsilon_{0}>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0})$ there exists a compact connected manifold
$S_{\varepsilon}$ which is

  1. (i)

$O(\varepsilon)-$close and diffeomorphic to $S$;

  2. (ii)

$C^{r}-$smooth in both $z$ and $\varepsilon$;

  3. (iii)

locally invariant under the map (36), i.e. the restricted map
$H|_{S_{\varepsilon}}$ is invertible and satisfies the following: if $z\in
S_{\varepsilon}$ and $H^{j}(z,\varepsilon)\in\mathcal{U}$ for all
$j=1,\ldots,l$, then

$\bar{z}^{j}=H^{j}(z,\varepsilon)\in S_{\varepsilon}$

for all $j=1,\ldots,l$, and if $z\in S_{\varepsilon}$ and
$H^{-j}(z,\varepsilon)\in\mathcal{U}$ for all $j=1,\ldots,l$, then

$\bar{z}^{-j}=H^{-j}(z,\varepsilon)\in S_{\varepsilon}$

for all $j=1,\ldots,l$.

The situation is sketched in Figure 6. Note that Theorem 3.1 also justifies
the key assumption for the derivation of the reduced map (24) via Proposition
2.14, namely, the existence of a locally invariant slow manifold
$S_{\varepsilon}$ perturbing from $S$. Taken together, Theorem 3.1 implies the
existence of a locally invariant slow manifold $S_{\varepsilon}$ that is
$O(\varepsilon)-$close to $S$, and Proposition 2.14 provides an explicit form
for the map governing the dynamics on $S_{\varepsilon}$.

It is also worthy to note that in general, slow manifolds described by Theorem
3.1 are non-unique. This fact, which is also true for Fenichel slow manifolds
in the ODE setting and for center-type manifolds more generally [17, 34, 42],
follows from the use of $C^{\infty}$ cutoff functions in order to control the
dynamics entering or leaving a neighbourhood of $S$ along the center
directions. As a consequence, any two slow (resp. Fenichel, center) manifolds
must have the same Taylor series in $\varepsilon$, however they may still be
separated by $O(e^{-c/\varepsilon})$ distances beyond all orders, where $c>0$.
Because of the closeness of slow manifolds, we shall frequently refer to any
fixed choice of slow manifold $S_{\varepsilon}$ described by Theorem 3.1 as
‘the’ slow manifold, as has become common nomenclature in fast-slow ODE
theory.

Figure 6: A normally hyperbolic critical manifold $S$ for the map (36)
perturbs to an $O(\varepsilon)-$close locally invariant slow manifold
$S_{\varepsilon}$ for $0<\varepsilon\ll 1$. This is described in Theorem 3.1.
Dynamics within $S_{\varepsilon}$ are governed by the map (21), i.e.
approximated to leading order in $\varepsilon$ by the reduced map (24).
Several iterates
$\bar{z}_{\varepsilon},\bar{z}_{\varepsilon}^{2},\ldots,\bar{z}_{\varepsilon}^{6}\in
S_{\varepsilon}$ starting from a point $z_{\varepsilon}\in S_{\varepsilon}$
are shown for illustrative purposes. Theorem 3.2 provides further details for
the case in which $S$ is has a graph representation $y=\varphi_{0}(x)$, in
which case $S_{\varepsilon}$ can be calculated up to $O(\varepsilon^{2})$
using the formula (38).

In order to provide a graph formulation of Theorem 3.1, we impose an
additional assumption.

######  Assumption 3.

The critical manifold $S$ is given as a graph of a $C^{r}-$smooth function
$\varphi_{0}:K\to\mathbb{R}^{k}$ over a compact, simply connected domain
$\mathcal{K}\subset\mathbb{R}^{n-k}$, the boundary of which is a $C^{\infty}$
$(k-1)-$dimensional submanifold $\partial\mathcal{K}$. We write

$S=\left\\{(x,\varphi_{0}(x)):x\in\mathcal{K}\right\\}.$

Assumption 3 is actually quite natural, in the sense that it can always be
satisfied locally after a change of notation if necessary since the matrix
$Df|_{S}$ has full row rank by Assumption 1. In order to consider maps (36)
satisfying Assumption 3, we reintroduce the componentwise notation used
already in the proofs of Propositions 2.8, 2.14, 2.19 and 2.22, i.e. we write
(36) as

$\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto\begin{pmatrix}\bar{x}\\\
\bar{y}\end{pmatrix}=\begin{pmatrix}H^{x}(x,y,\varepsilon)\\\
H^{y}(x,y,\varepsilon)\end{pmatrix}=\begin{pmatrix}x\\\
y\end{pmatrix}+\begin{pmatrix}N^{x}(x,y)\\\
N^{y}(x,y)\end{pmatrix}f(x,y)+\varepsilon\begin{pmatrix}G^{x}(x,y,\varepsilon)\\\
G^{y}(x,y,\varepsilon)\end{pmatrix},$ (37)

where as before $x\in\mathbb{R}^{k}$, $y\in\mathbb{R}^{n-k}$, $N^{x}(x,y)$ and
$N^{y}(x,y)$ are matrices of dimensions $k\times(n-k)$ respectively
$(n-k)\times(n-k)$, and $G^{x}(x,y,\varepsilon)$, $G^{y}(x,y,\varepsilon)$ are
column vectors of length $k$ respectively $n-k$, and $f(x,y)=y-\varphi_{0}(x)$
after redefining $N(x,y)$ if necessary.

######  Theorem 3.2.

(Existence of slow manifolds as graphs) Consider the map (37) under
Assumptions 1, 2 and 3. Then there exists a $C^{r}-$smooth function
$\varphi_{\varepsilon}:\mathcal{K}\times[0,\varepsilon_{0})\to\mathbb{R}^{k}$
such that the slow manifold is also a graph

$S_{\varepsilon}=\left\\{(x,\varphi_{\varepsilon}(x)):x\in\mathcal{K}\right\\},$

where in particular

$\varphi_{\varepsilon}(x)=\varphi_{0}(x)-\varepsilon(D_{y}f)^{-1}(DfN)^{-1}DfG+O(\varepsilon^{2}).$
(38)

Moreover, $S_{\varepsilon}$ is locally invariant in the following sense: for
each $(x,y)\in S_{\varepsilon}$ the invariance equation

$\bar{y}=\varphi_{\varepsilon}\left(H^{x}(x,\varphi_{\varepsilon}(x),\varepsilon)\right)=H^{y}\left(x,\varphi_{\varepsilon}(x),\varepsilon\right)$
(39)

is satisfied as long as $H(x,y,\varepsilon)\in\mathcal{U}$, and the invariance
equation

$\bar{y}^{-1}=\varphi_{\varepsilon}\left((H^{x})^{-1}(x,\varphi_{\varepsilon}(x),\varepsilon)\right)=(H^{y})^{-1}\left(x,\varphi_{\varepsilon}(x),\varepsilon\right)$
(40)

is satisfied as long as $H^{-1}(x,y,\varepsilon)\in\mathcal{U}$.

Theorem 3.1 can be shown to follow from the graph formulation in Theorem 3.2
using compactness and a partition of unity. Hence in Section 4 it is
sufficient to prove Theorem 3.2. Slow manifold properties such as smoothness
follow from the existence of the $C^{r}-$smooth function
$\varphi_{\varepsilon}$. Equation (38) in Theorem 3.2 provides the asymptotics
for the slow manifold $S_{\varepsilon}$. In particular, this expression
constitutes a local formula for $S_{\varepsilon}$ up to $O(\varepsilon^{2})$
in terms of the initial data $N$, $f$ and $G$.

We turn now to persistence of stable and unstable manifolds $W_{loc}^{s}(S)$
and $W_{loc}^{u}(S)$ respectively, as defined for $\varepsilon=0$ in (19).
Similarly to the case of fast-slow ODEs, both stable and unstable manifolds
can be shown to perturb to nearby manifolds with certain local invariance
properties.

######  Theorem 3.3.

(Persistence of stable/unstable manifolds) Consider the map (36) under
Assumptions 1-2. Then for all $\varepsilon\in(0,\varepsilon_{0})$ with
$\varepsilon_{0}>0$ sufficiently small there exists manifolds
$W_{loc}^{s}(S_{\varepsilon})$ and $W_{loc}^{u}(S_{\varepsilon})$ that are

  1. (i)

$O(\varepsilon)-$close and diffeomorphic to $W_{loc}^{s}(S)$ and
$W_{loc}^{u}(S)$ respectively;

  2. (ii)

$C^{r}-$smooth in both $z$ and $\varepsilon$;

  3. (iii)

locally positively and negatively invariant under the map (10), respectively.
More precisely, the restricted map $H|_{W^{u}_{loc}(S_{\varepsilon})}$ is
invertible and satisfies the following: if $z\in W_{loc}^{s}(S_{\varepsilon})$
and $H^{j}(z,\varepsilon)\in\mathcal{U}$ for all $j=1,\ldots,l$, then

$\bar{z}^{j}=H^{j}(z,\varepsilon)\in W_{loc}^{s}(S_{\varepsilon})$

for all $j=1,\ldots,l$. Similarly, if $z\in W_{loc}^{u}(S_{\varepsilon})$ and
$H^{-j}(z,\varepsilon)\in\mathcal{U}$ for all $j=1,\ldots,l$, then

$\bar{z}^{-j}=H^{-j}(z,\varepsilon)\in W_{loc}^{u}(S_{\varepsilon}),$

for all $j=1,\ldots,l$.

Moreover, the slow manifold $S_{\varepsilon}$ lies within the intersection of
the perturbed stable and unstable manifolds, i.e.

$S_{\varepsilon}=W_{loc}^{s}(S_{\varepsilon})\cap
W_{loc}^{u}(S_{\varepsilon}).$

The situation is sketched in Figure 7. The persistence of stable and unstable
manifolds $W_{loc}^{s/u}(S)$ as nearby locally invariant manifolds
$W_{loc}^{s/u}(S_{\varepsilon})$ described by Theorem 3.3 is essentially
analogous to the persistence of normally hyperbolic critical manifolds $S$ as
nearby locally invariant slow manifolds $S_{\varepsilon}$ in Theorem 3.1. In
particular (and for similar reasons), the manifolds
$W_{loc}^{s/u}(S_{\varepsilon})$ are also non-unique but
$O(e^{-c/\varepsilon})-$close for some constant $c>0$. The final assertion
that $S_{\varepsilon}$ lies in the intersection
$W_{loc}^{s}(S_{\varepsilon})\cap W_{loc}^{u}(S_{\varepsilon})$ explains the
similarities between Theorems 3.1 and 3.3. In fact, as is typical in proofs
for the existence of normally hyperbolic manifolds more generally, Theorem 3.1
will be derived (via Theorem 3.2) in Section 4 as a direct consequence of
Theorem 3.3.

Figure 7: The stable/unstable manifolds $W^{u/s}_{loc}(S)$ (here in shaded
blue/red) of a normally hyperbolic critical manifold $S$ (shaded green)
perturb to $O(\varepsilon)-$close positively/negatively invariant manifolds
$W^{s/u}_{loc}(S_{\varepsilon})$ (blue/red) which intersect along the slow
manifold $S_{\varepsilon}$ (green), see Theorem 3.3.

Finally and similarly to the case for ODEs, it turns out the manifolds
$W_{loc}^{s/u}(S_{\varepsilon})$ admit an invariant foliation by smooth fibers
with base points on $S_{\varepsilon}$, i.e. the invariant foliation of the
manifolds $W_{loc}^{u/s}(S)$ by unstable/stable manifolds of points on $S$,
see again (42), also persist. The family of stable/unstable fibers also
exhibit useful invariance properties, and a quantitative estimate for the
contraction/expansion rate along fibers can be given in terms of spectral
information.

We first require a little more notation. Recall from Definition 2.6 that the
non-trivial multipliers of a $k-$dimensional normally hyperbolic critical
manifold $S$ are denoted by $\mu_{j}(z)$, where $z\in S$ and $j=1,\ldots,n-k$.
By normal hyperbolicity, $|\mu_{j}(z)|\neq 1$ for all $j$. Denote the
$n_{a}\leq n-k$ stable multipliers with $|\mu_{j}(z)|<1$ by $\mu_{a,j}(z)$,
and the $n_{r}\leq n-k$ unstable multipliers with $|\mu_{j}(z)|>1$ by
$\mu_{r,j}(z)$, where $n_{a}+n_{r}=n-k$. Assuming that $S$ is compact, we may
define spectral bounds

$\nu_{A}:=\sup_{z\in
S,j=1,\ldots,n_{a}}|\mu_{a,j}(z)|<1,\qquad\nu_{R}:=\inf_{z\in
S,j=1,\ldots,n_{r}}|\mu_{r,j}(z)|>1.$ (41)

Contraction and repulsion along stable and unstable fibers respectively can be
quantified in terms of $\nu_{A}$ and $\nu_{R}$. We now state the main result
on the persistence of stable and unstable foliations.

(a) Orientation preserving case.

(b) Orientation reversing case.

Figure 8: Positive invariance of the perturbed stable fibers
$\\{w_{loc}^{s}(z_{\varepsilon})\\}_{z_{\varepsilon}\in S_{\varepsilon}}$ of a
$1-$dimensional critical manifold in dimension $n=2$. By the invariance
property in Theorem 3.4 (i), initial conditions $q_{1},q_{2}$ in the stable
fiber $w^{s}_{loc}(z_{\varepsilon,1})$ with base point $z_{\varepsilon,1}\in
S_{\varepsilon}$ are mapped by $H$ into the stable fiber
$w^{s}_{loc}(z_{\varepsilon,2})$, then $w^{s}_{loc}(z_{\varepsilon,3})$, with
base points $z_{\varepsilon,2}$ and $z_{\varepsilon,3}$ corresponding to
iterates of $z_{\varepsilon,1}$. Moreover, iterates are exponentially
contracted towards their base points on $S_{\varepsilon}$ by Theorem 3.4
(iii). In (a): the orientation preserving case in which $S$ has a non-trivial
multiplier $\mu\in(0,1)$. In (b): the orientation reversing case with
$\mu\in(-1,0)$.

######  Theorem 3.4.

(Persistence of stable/unstable foliations) Assume the same hypotheses as
Theorem 3.3. Then for $\varepsilon\in(0,\varepsilon_{0})$ with
$\varepsilon_{0}>0$ sufficiently small, the perturbed stable/unstable
manifolds $W^{s/u}_{loc}(S_{\varepsilon})$ of Theorem 3.3 admit foliations by
stable/unstable fibers $w^{s/u}_{loc}(z_{\varepsilon})$ with base points
$z_{\varepsilon}\in S_{\varepsilon}$, i.e.

$W^{s}_{loc}(S_{\varepsilon})=\bigcup_{z_{\varepsilon}\in
S_{\varepsilon}}w^{s}_{loc}(z_{\varepsilon}),\qquad
W^{u}_{loc}(S_{\varepsilon})=\bigcup_{z_{\varepsilon}\in
S_{\varepsilon}}w^{u}_{loc}(z_{\varepsilon}).$ (42)

These foliations converge fiberwise to the foliations (19) in the layer map as
$\varepsilon\to 0$, and satisfy the following:

  1. (i)

The stable fibers $w^{s}_{loc}(z_{\varepsilon})$ form a locally positively
invariant family, i.e. if $z_{\varepsilon}\in S_{\varepsilon}$ and
$H(z_{\varepsilon},\varepsilon)\in\mathcal{U}$, then

$H\left(w_{loc}^{s}(z_{\varepsilon}),\varepsilon\right)\subset
w_{loc}^{s}\left(H(z_{\varepsilon},\varepsilon)\right).$

  2. (ii)

The unstable fibers $w^{u}_{loc}(z_{\varepsilon})$ form a locally negatively
invariant family, i.e. if $z_{\varepsilon}\in S_{\varepsilon}$ and
$H^{-1}(z_{\varepsilon},\varepsilon)\in\mathcal{U}$, then

$H^{-1}\left(w_{loc}^{u}(z_{\varepsilon}),\varepsilon\right)\subset
w_{loc}^{u}\left(H^{-1}(z_{\varepsilon},\varepsilon)\right).$

  3. (iii)

For each $z_{\varepsilon}\in S_{\varepsilon}$ such that
$H(z_{\varepsilon},\varepsilon)\in\mathcal{U}$ for all $j=1,\ldots,l$,
contraction along stable fibers $w_{loc}^{s}(z_{\varepsilon})$ is exponential
with rate faster than $\chi_{A}^{j}$, for a fixed constant
$\chi_{A}\in(\nu_{A},1)$. More precisely, for all $\xi_{\varepsilon}\in
w_{loc}^{s}(z_{\varepsilon})$ such that $H^{j}(\xi_{\varepsilon},\varepsilon)$
and $H^{j}(z_{\varepsilon},\varepsilon)$ stay in $\mathcal{U}$ for all
$j=1,\ldots,l$, we have that

$\|H^{j}(\xi_{\varepsilon},\varepsilon)-H^{j}(z_{\varepsilon},\varepsilon)\|\leq\chi_{A}^{j}\|\xi_{\varepsilon}-z_{\varepsilon}\|.$

  4. (iv)

For each $z_{\varepsilon}\in S_{\varepsilon}$ such that
$H^{-1}(z_{\varepsilon},\varepsilon)\in\mathcal{U}$ for all $j=1,\ldots,l$,
contraction along unstable fibers $w_{loc}^{s}(z_{\varepsilon})$ is
exponential with rate faster than $\chi_{R}^{-j}$, for a fixed constant
$\chi_{R}\in(1,\nu_{R})$. More precisely, for all $\xi_{\varepsilon}\in
w_{loc}^{u}(z_{\varepsilon})$ such that
$H^{-j}(\xi_{\varepsilon},\varepsilon)$ and
$H^{-j}(z_{\varepsilon},\varepsilon)$ stay in $\mathcal{U}$ for all
$j=1,\ldots,l$, we have that

$\|H^{-j}(\xi_{\varepsilon},\varepsilon)-H^{-j}(z_{\varepsilon},\varepsilon)\|\leq\chi_{R}^{-j}\|\xi_{\varepsilon}-z_{\varepsilon}\|.$

  5. (v)

The stable fibers $w^{s}_{loc}(z_{\varepsilon})$ are Lipschitz manifolds. If
additionally degenerate superstability in the sense that

$\mu_{a,j}(z)=0$

for all $j=1,\ldots,n_{a}$ only occurs at isolated points $z\in S$, then
stable fibers $w^{s}_{loc}(z_{\varepsilon})$ are $C^{r-1}-$smooth.

  6. (vi)

The unstable fibers $w^{u}_{loc}(z_{\varepsilon})$ are $C^{r-1}-$smooth.

Theorem 3.4 asserts that the foliation of stable/unstable manifolds
$W_{loc}^{s/u}(S)$ by lower dimensional stable/unstable manifolds
$w^{s/u}_{loc}(z)$ associated to points $z\in S$, also perturb in a regular
fashion. The local positive/negative invariance of the families
$\\{w_{loc}^{s/u}(z_{\varepsilon})\\}_{z_{\varepsilon}\in S_{\varepsilon}}$ is
described in assertions (i)-(ii), and sketched for the stable foliation in
Figure 8. Statements (iii)-(iv) assert the exponential contraction and
repulsion along stable and unstable fibers respectively. This explains the
common use of terms like “stable”, “unstable” and “normally hyperbolic” when
describing perturbed fibers or manifolds like
$w_{loc}^{s/u}(z_{\varepsilon})$, $W^{s/u}_{loc}(S_{\varepsilon})$ and
$S_{\varepsilon}$. Notice by assertions (v)-(vi) that stable/unstable fibers
$w_{loc}^{s/u}(z_{\varepsilon})$ are only $C^{r-1}-$smooth with respect to
variation of the base point $z_{\varepsilon}\in S_{\varepsilon}$, even though
by Theorem 3.3 the stable/unstable manifolds $W^{s/u}_{loc}(S_{\varepsilon})$
themselves are $C^{r}-$smooth. This is also true of foliations with Fenichel
slow manifolds as base in the ODE setting [17, 34, 72, 73]. In contrast to the
ODE setting, however, an additional nondegeneracy condition is required in
assertion (v) in order to infer $C^{r-1}-$smoothness of the stable fibers in
particular.

######  Remark 3.5.

The proof of Theorem 3.4 assertion (v) in Section 4.3.3 relies on a smoothness
result from [54] which cannot be applied in the (very) degenerate superstable
case in which the critical manifold $S$ has a connected component $U_{ss}$
with $\mu_{a}(z)\equiv 0$ for $z\in S\cap U_{ss}$. The question of whether the
nondegeneracy in Theorem 3.4 is a necessary condition for smoothness of the
stable fibers is not considered further in this work.

##  4 Proofs

In the following we prove the main results from Section 3. Our results can be
viewed as a specialisations of pre-existing results on the existence of
normally hyperbolic manifolds and their foliations more generally.
Specifically in this work, the majority of our results will be derived as
specialisations of more general theorems in [54] to the case of fast-slow maps
in the general nonstandard form (36). In order to apply the results in [54],
the equations must first be ‘prepared’ to a certain extent, and herein lies
much of the analysis. The aim is to formulate necessary and sufficient
conditions in [54] in terms of coordinate-invariant properties of the map,
e.g. the multipliers along $S$, so that results obtained for normal forms can
be directly related to the original map (36). Further work must be done to
prove the local invariance of slow manifolds and foliations described by our
results, since the results in [54] and related works rely on some simplifying
assumptions which yield only to globally invariant objects. Finally,
additional work is needed in order to derive those features which are
characteristic of or specific to fast-slow maps, for example, calculations
leading to direct estimates for the slow manifold parameterization, or for the
asymptotic decay rates along invariant fibers.

This section is structured as follows: In Section 4.1 we introduce notation,
identify suitable neighbourhoods and derive local coordinates such that the
relevant inflowing/outflowing requirements for the application of results in
[54] are satisfied. In Section 4.2 we verify (a suitable reformulation of)
necessary and sufficient conditions for the application of these results.
Finally in Section 4.3, we apply the results from [54] and use them to prove
our main results from Section 3.

###  4.1 Preparatory results

We begin by identifying neighbourhoods and local coordinates suited to our
needs. The main task is to separate directions along which iterates of (36)
are inflowing and outflowing respectively.

For local analyses we can assume the map (36) is in the form (37). Recall that
this form is always achievable locally, i.e. does not rely on Assumption 3.
The first simplification is to rectify $S$ along the $x-$axes via the
transformation $v=f(x,y)$, which has a locally defined inverse $y=K(x,v)$. As
shown in the proof of Proposition 2.8, this leads to

$\begin{split}x&\mapsto\bar{x}=x+\tilde{N}^{x}(x,v)v+\varepsilon\tilde{G}^{x}(x,v,\varepsilon),\\\
v&\mapsto\bar{v}=v+DfN(x,K(x,v))v+\varepsilon
DfG(x,K(x,v),\varepsilon)+O(v^{2},v\varepsilon,\varepsilon^{2}),\end{split}$
(43)

where we have defined $\tilde{N}^{x}(x,v):=N^{x}(x,K(x,v))$,
$\tilde{G}^{x}(x,v,\varepsilon):=G^{x}(x,K(x,v),\varepsilon)$, and
$Df=(D_{x}f,D_{y}f)$. In these coordinates the critical manifold is simply

$S=\\{(x,0):x\in\mathcal{K}\\},$

where we may (and will) assume that $\mathcal{K}\subset\mathbb{R}^{k}$ is
compact, simply connected, and that $S$ is normally hyperbolic. Fast
directions are encoded in (but not identified with) the variables
$v\in\mathbb{R}^{n-k}$, since the $n-k$ multipliers of the linear part
$I_{n-k}+DfN|_{S}$ coincide by Proposition 2.8 with the $n-k$ non-trivial
multipliers $\mu_{j}$ associated with the fast directions.

######  Lemma 4.1.

Fix $\varepsilon_{0}>0$ sufficiently small. For all
$\varepsilon\in[0,\varepsilon_{0})$, there exists a local coordinate
transformation transforming (43) into

$\begin{split}x&\mapsto\bar{x}=x+\tilde{N}^{x}(x,N^{y}(a,b)^{\textnormal{T}})N^{y}(a,b)^{\textnormal{T}}+\varepsilon\tilde{G}^{x}(x,N^{y}(a,b)^{\textnormal{T}},\varepsilon),\\\
a&\mapsto\bar{a}=a+\Lambda_{a}(x,a,b)a+\varepsilon(N_{a}^{y})^{-1}DfG(x,N^{y}(a,b)^{\textnormal{T}},\varepsilon),\\\
b&\mapsto\bar{b}=b+\Lambda_{r}(x,a,b)b+\varepsilon(N_{r}^{y})^{-1}DfG(x,N^{y}(a,b)^{\textnormal{T}},\varepsilon),\end{split}$
(44)

where $\Lambda_{a}(x,0,0)$ is a diagonal $n_{a}\times n_{a}$ matrix with
eigenvalues $\lambda_{i}(x)$ such that

$|\mu_{i}(x)|=|1+\lambda_{i}(x)|<1$

uniformly in $x$ for all $i=1,\ldots,n_{a}$, and $\Lambda_{r}(x,0,0)$ is a
diagonal $n_{r}\times n_{r}$ matrix with eigenvalues $\lambda_{i}(x)$ such
that

$|\mu_{i}(x)|=|1+\lambda_{i}(x)|>1$

uniformly in $x$ for all $i=1,\ldots,n_{r}$.

###### Proof.

We use the following facts:

  * •

For any $z\in S$,

$\textrm{Im}\,Dh(z)=\mathcal{N}_{z}=E^{s}(z)\cup
E^{u}(z)=\textrm{span\,}\\{N^{i}(z):i=1,\ldots,n-k\\},$

where the $N^{i}(z)$ are columns of $N(z)$.

  * •

For any $z\in S$,

$\ker Dh(z)=T_{z}S=\textrm{span\,}\\{Df_{i}(z):i=1,\ldots,n-k\\}^{\perp},$

where the $Df_{i}(z)$ are the rows of $Df(z)$, and $\perp$ denotes the
orthogonal complement.

It follows from these facts that for each $z\in S$, the $n\times n$ matrix
$A(z):=[Df(z)^{\perp}\ N(z)]$ forms the matrix with eigenvector columns needed
to make a Jordan decomposition of the layer map (43)$|_{\varepsilon=0}$. Here
is suffices to Jordan decompose only the $v$ variables which, in the case of
saddle-type critical manifolds, will allow for a splitting of attracting and
repelling components in the leading order. Since column vectors of $N(z)$ can
be chosen to be eigenvectors spanning $E^{u}(z)\cup E^{s}(z)$ at each $z\in
S$, the (linearly independent) column vectors of the $(n-k)\times(n-k)$ matrix
$N^{y}(z)$ are eigenvectors of the matrix $DfN(z)$. This motivates the
coordinate transformation

$u=(N^{y})^{-1}v\in\mathbb{R}^{n-k},$

which leads to the map

$\begin{split}x&\mapsto\bar{x}=x+\tilde{N}^{x}(x,N^{y}u)N^{y}u+\varepsilon\tilde{G}^{x}(x,N^{y}u,\varepsilon),\\\
u&\mapsto\bar{u}=u+\Lambda(x,u)u+\varepsilon(N^{y})^{-1}DfG(x,N^{y}u,\varepsilon)+O(u^{2},\varepsilon
u,\varepsilon^{2}),\end{split}$

where
$I_{n-k}+\Lambda(x,0)=\textrm{diag}\\{\mu_{1}(x),\ldots,\mu_{n-k}(x)\\}$.
Since it is achievable by a simple permutation of notation if necessary, we
may assume without loss of generality that $|\mu_{j}(x)|<1$ for
$j=1,\ldots,n_{a}$ and $|\mu_{j}(x)|>1$ for the remaining $n_{r}$ multipliers
with $j=n_{a}+1,\ldots,n-k$ (recall that $n_{a}+n_{r}=n-k$). In this case the
matrix $\Lambda$ has a block-diagonal structure, and we may write

$\begin{split}x&\mapsto\bar{x}=x+\tilde{N}^{x}(x,N^{y}u)N^{y}u+\varepsilon\tilde{G}^{x}(x,N^{y}u,\varepsilon),\\\
a&\mapsto\bar{a}=a+\Lambda_{a}(x,u)a+\varepsilon(N_{a}^{y})^{-1}DfG(x,N^{y}u,\varepsilon),\\\
b&\mapsto\bar{b}=b+\Lambda_{r}(x,u)b+\varepsilon(N_{r}^{y})^{-1}DfG(x,N^{y}u,\varepsilon),\\\
\end{split}$

for suitably defined $N^{y}_{a}$, $N^{y}_{r}$, where
$u=(a,b)^{\textnormal{T}}\in\mathbb{R}^{n_{a}}\times\mathbb{R}^{n_{r}}$, and
the diagonal matrices $I_{n_{a}}+\Lambda_{a}(x,0)$ and
$I_{n_{r}}+\Lambda_{r}(x,0)$ encode all and only the attracting and repelling
multipliers respectively. ∎

In order to meet the relevant inflowing/outflowing and invariance criteria,
the set $\mathcal{K}$ needs to be extended. Specifically, results in [54]
apply for open neighbourhoods $X\times Y$ for which $X$ is inflowing and $Y$
is outflowing (or visa-versa) with respect to the map. As in all proofs of the
center manifold theorem, escape along the slow (i.e. center) directions poses
a problem for the satisfaction of certain global invariance properties. The
usual approach to controlling the slow directions is to enlarge the
neighbourhood $\mathcal{K}$, and augment the slow dynamics by the addition of
a suitable cutoff/bump function which prevents iterates from escaping this
larger neighbourhood. Results in [54] which only apply for neighbourhoods with
global inflowing/outflowing and invariance properties can be applied on this
enlarged neighbourhood, and subsequently restricted to $\mathcal{K}$.

The existence of a suitable enlargement $\hat{\mathcal{K}}$ such that
$\mathcal{K}\subset\text{Int\ }\hat{\mathcal{K}}$ is guaranteed by smoothness.
In particular, following the analogous setup for ODEs in [17, 34],
$\hat{\mathcal{K}}$ can be chosen so that $S$ is defined/extended over
$\hat{\mathcal{K}}$ as a $C^{r}-$smooth graph

$\hat{S}=\left\\{(x,\phi_{0}(x)):x\in\hat{\mathcal{K}}\right\\},$ (45)

such that $\phi_{0}|_{\mathcal{K}}(x)\equiv 0$, with boundary
$\partial\hat{\mathcal{K}}$ given by $\hat{\omega}(x)=0$ for a
$C^{\infty}-$smooth function $\hat{\omega}$ such that
$\nabla\hat{\omega}|_{\partial\hat{\mathcal{K}}}\neq 0$. In fact, let
$\hat{\omega}(x)$ be normalised so that $n_{x}:=\nabla\hat{\omega}(x)$ defines
an outward pointing normal to $\partial\hat{\mathcal{K}}$. Now consider the
map

$\begin{split}x&\mapsto\bar{x}=x+\tilde{N}^{x}(x,N^{y}u)N^{y}(a,b)^{\textnormal{T}}+\varepsilon\tilde{G}^{x}(x,N^{y}(a,b)^{\textnormal{T}},\varepsilon)+\hat{\delta}\rho(x)n_{x},\\\
a&\mapsto\bar{a}=a+\Lambda_{a}(x,a,b)a+\varepsilon(N_{a}^{y})^{-1}DfG(x,N^{y}(a,b)^{\textnormal{T}},\varepsilon),\\\
b&\mapsto\bar{b}=b+\Lambda_{r}(x,a,b)b+\varepsilon(N_{r}^{y})^{-1}DfG(x,N^{y}(a,b)^{\textnormal{T}},\varepsilon),\end{split}$
(46)

where $\hat{\delta}>0$ is a constant to be specified later on, and $\rho(x)$
is a $C^{\infty}$ function contributing only on
$\mathbb{R}^{k}\setminus\hat{\mathcal{K}}$, i.e.

$\rho(x)=\begin{cases}1,&x\in\mathbb{R}^{k}\setminus\hat{\mathcal{K}},\\\
0,&x\in\mathcal{K},\end{cases}$

and $\rho(x)\in(0,1)$ for $x\in\hat{\mathcal{K}}\setminus\mathcal{K}$. Due to
compactness, such a function can always be constructed using local
$C^{\infty}$ bump functions and a partition of unity. It is important to note
that the maps (44) and (46) agree on $\mathcal{K}$.

Finally, it will be helpful to have a notation for spectral bounds. We define

$\nu_{A}:=\sup_{x\in\hat{\mathcal{K}},j=1,\ldots,n_{a}}|\mu_{a,j}(x)|<1,\qquad\nu_{R}:=\inf_{x\in\hat{\mathcal{K}},j=1,\ldots,n_{r}}|\mu_{r,j}(x)|>1,$
(47)

where $\mu_{a,j}(x)$ and $\mu_{r,j}(x)$ denote multipliers of the matrices
$I_{n_{a}}+\Lambda_{a}|_{\hat{S}}$ and $I_{n_{r}}+\Lambda_{r}|_{\hat{S}}$
respectively. Note that $\nu_{A}$ and $\nu_{R}$ agree with their previous
definition in (41) since the multipliers $\mu_{a,j}$ and $\mu_{r,j}$ are
invariant under $C^{r}$ coordinate transformations.

###  4.2 Checking necessary and sufficient conditions

We now show that the relevant invariance properties for the application of
results in [54] are satisfied for the map (46). This will be easier with a
little extra notation. We first rewrite the map (46) as

$\begin{split}b&\mapsto\bar{b}=\widehat{H}^{b}(b,x,a,\varepsilon),\\\
x&\mapsto\bar{x}=\widehat{H}^{x}(b,x,a,\varepsilon),\\\
a&\mapsto\bar{a}=\widehat{H}^{a}(b,x,a,\varepsilon),\end{split}$ (48)

where in particular, we have permuted the order of the equations and the
arguments in order for simplicity in the notation to follow. Our analysis here
is purely local, and may be restricted to

$b\in B:=\\{b\in\mathbb{R}^{n_{r}}:|b|\leq\delta_{0}\\},\quad a\in
A:=\\{a\in\mathbb{R}^{n_{a}}:|a|\leq\delta_{0}\\},\quad\varepsilon\in[0,\varepsilon_{0}),$

for fixed $\delta_{0},\varepsilon_{0}>0$ which we shall frequently choose to
be sufficiently small for the validity of estimates. We are also free to
choose $\mathcal{K}$ and $\hat{\mathcal{K}}$ in such a way that

$x\in\mathcal{K}\ \implies\ |x|<\Delta,\qquad x\in\hat{\mathcal{K}}\ \implies\
|x|<\hat{\Delta},$

for fixed $\hat{\Delta}>\Delta>0$.

Following [54], we now introduce two more equivalent formulations that are
better suited for the analyses of attracting and repelling invariant objects
respectively. Consider the attracting formulation first. In this case, we
write

$x_{A}:=(b,x)^{\textnormal{T}}\in X_{A},\qquad y_{A}=a\in Y_{A},$

define

$\widehat{H}_{A}^{x_{A}}(x_{A},y_{A},\varepsilon):=\begin{pmatrix}\widehat{H}^{b}(x_{A},y_{A},\varepsilon)\\\
\widehat{H}^{x}(x_{A},y_{A},\varepsilon)\end{pmatrix},\qquad\widehat{H}^{y_{A}}_{A}(x_{A},y_{A},\varepsilon):=\widehat{H}^{a}(x_{A},y_{A},\varepsilon),$

and consider the map (48) expressed as

$\widehat{H}_{A}:\begin{pmatrix}x_{A}\\\
y_{A}\end{pmatrix}\mapsto\begin{pmatrix}\bar{x}_{A}\\\
\bar{y}_{A}\end{pmatrix}=\begin{pmatrix}\widehat{H}_{A}^{x_{A}}(x_{A},y_{A},\varepsilon)\\\
\widehat{H}_{A}^{y_{A}}(x_{A},y_{A},\varepsilon)\end{pmatrix}.$ (49)

For the repelling formulation, we write

$x_{R}:=b\in X_{R},\qquad y_{R}:=(x,a)^{\textnormal{T}}\in Y_{R},$

define

$\widehat{H}_{R}^{x_{R}}(x_{R},y_{R},\varepsilon):=\widehat{H}^{b}(x_{R},y_{R},\varepsilon),\qquad\widehat{H}_{R}^{y_{R}}(x_{R},y_{R},\varepsilon):=\begin{pmatrix}\widehat{H}^{x}(x_{R},y_{R},\varepsilon)\\\
\widehat{H}^{a}(x_{R},y_{R},\varepsilon)\end{pmatrix},$

and consider the map (48) expressed as

$\widehat{H}_{R}:\begin{pmatrix}x_{R}\\\
y_{R}\end{pmatrix}\mapsto\begin{pmatrix}\bar{x}_{R}\\\
\bar{y}_{R}\end{pmatrix}=\begin{pmatrix}\widehat{H}_{R}^{x_{R}}(x_{R},y_{R},\varepsilon)\\\
\widehat{H}_{R}^{y_{R}}(x_{R},y_{R},\varepsilon)\end{pmatrix}.$ (50)

####  4.2.1 Invariance conditions

The relevant inflowing, outflowing and invariance conditions are summarized in
the following result.

######  Lemma 4.2.

(c.f. [54, Hypothesis HM]) For each

$0<\hat{\delta}<\min\left\\{\frac{\hat{\Delta}-(\Delta+c_{0})}{C},\frac{\Delta^{2}}{\hat{\Delta}(1+M\hat{\Delta})}\right\\},$
(51)

where $M:=\sup_{x\in\hat{\mathcal{K}}}|x\hat{\rho}^{\prime}(x)|$, there exists
$\delta_{0},\varepsilon_{0}>0$ such that the map (48) has the following
invariance properties for all $\varepsilon\in[0,\varepsilon_{0})$:

  1. 1.

$\widehat{H}_{A}$ is inflowing with respect to $Y_{A}$, i.e.

$(x_{A},y_{A})\in X_{A}\times
Y_{A}\quad\implies\quad\widehat{H}^{y_{A}}_{A}(x_{A},y_{A},\varepsilon)\in
Y_{A},$

and outflowing with respect to $X_{A}$, i.e. for all $(\bar{x}_{A},y_{A})\in
X_{A}\times Y_{A}$ there exists an $x_{A}\in X_{A}$ such that

$\bar{x}_{A}=\widehat{H}_{A}^{x_{A}}(x_{A},y_{A},\varepsilon).$

  2. 2.

$\widehat{H}_{R}$ is inflowing with respect to $Y_{R}$, i.e.

$(x_{R},y_{R})\in X_{R}\times
Y_{R}\quad\implies\quad\widehat{H}^{y_{R}}_{A}(x_{R},y_{R},\varepsilon)\in
Y_{R},$

and outflowing with respect to $X_{R}$, i.e. for all $(\bar{x}_{R},y_{R})\in
X_{R}\times Y_{R}$ there exists an $x_{R}\in X_{R}$ such that

$\bar{x}_{R}=\widehat{H}_{R}^{x_{R}}(x_{R},y_{R},\varepsilon).$

###### Proof.

We start with the statement (a). Since $\widehat{H}_{A}$ is smooth in all
arguments and we have the linear contractivity property

$\sup_{x\in\mathcal{K}}\left\lVert
I_{n_{a}}+\Lambda_{a}|_{\hat{S}}\right\rVert=\nu_{A}<1,$

we obtain the estimate

$|\widehat{H}_{A}^{x_{A}}(x_{A},y_{A},\varepsilon)|=|\widehat{H}^{a}(x_{A},y_{R},\varepsilon)|\leq\nu_{A}\delta_{0}+\delta_{0}^{2}c_{1}+\varepsilon_{0}c_{2}<\delta_{0}\\\
$ (52)

for all $(x_{A},y_{A})\in X_{A}\times Y_{A}$, for some constants
$c_{1},c_{2}\geq 0$, and where the rightmost inequality is obtained by
choosing $\delta_{0},\varepsilon_{0}>0$ sufficiently small. Hence
$\widehat{H}_{A}$ is inflowing with respect to $Y_{A}$.

In order to show that $\widehat{H}_{A}$ is outflowing with respect to $X_{A}$,
let $(\bar{x}_{A},y_{A})=((\bar{b},\bar{x}),a)\in X_{A}\times Y_{A}$ so that
in particular, $|\bar{x}|\leq\hat{\Delta}$ and $|\bar{b}|<\delta_{0}$. We need
to show that there exists $(x_{A},y_{A})=((b,x)^{\textnormal{T}},a)\in
X_{A}\times Y_{A}$ such that

$\bar{x}=\widehat{H}^{x}(b,x,a,\varepsilon)\qquad\text{and}\qquad\bar{b}=\widehat{H}^{b}(b,x,a,\varepsilon).$

The latter follows immediately, since the linear expansion property

$\inf_{x\in\hat{\mathcal{K}}}\left\lVert
I_{n_{r}}+\Lambda_{r}|_{\hat{S}}\right\rVert\geq\nu_{R}>1$

implies the existence of an inverse $(I_{n_{r}}+\Lambda_{r}(x,a,b))^{-1}$ for
$\delta_{0}>0$ sufficiently small. In particular,
$\bar{b}=\widehat{H}^{b}(b,x,a,\varepsilon)$ can be solved via the implicit
function theorem and the norm of $b$ can be estimated directly via the
implicit equation

$b=(I_{n_{r}}+\Lambda_{r}(x,a,b))^{-1}\bar{b}+O(\varepsilon),$ (53)

which satisfies $|b|<\delta_{0}$ as required for sufficiently small
$\delta_{0},\varepsilon_{0}>0$.

Now consider $\bar{x}=\widehat{H}^{x}(b,x,a,\varepsilon)$. We need to show
that this equation can be solved for an $x$ such that $x\in\hat{\mathcal{K}}$.
Towards this end, we define

$\hat{\rho}(x):=\rho(x)n_{x}\frac{|x|}{x}$

and consider

$x=A(x)^{-1}\left(\bar{x}-\tilde{N}^{x}(x,N^{y}u)N^{y}(a,b)^{\textnormal{T}}-\varepsilon\tilde{G}^{x}(x,N^{y}(a,b)^{\textnormal{T}},\varepsilon)\right)=:R(x),$

where

$A(x)^{-1}:=\left(1+\hat{\delta}\frac{\hat{\rho}(x)}{|x|}\right)^{-1}.$

Fixed points $R(x)=x$ correspond to solutions of
$\bar{x}=\widehat{H}^{x}(b,x,a,\varepsilon)$, and may be identified by a
contraction mapping argument if we can show that $R$ is a map
$R:\hat{\mathcal{K}}\to\hat{\mathcal{K}}$ satisfying $|R^{\prime}(x)|<1$ for
all $x\in\hat{\mathcal{K}}$. First, we show that
$R:\hat{\mathcal{K}}\to\hat{\mathcal{K}}$. Observe that

$|R(x)|\leq\frac{1}{1+\hat{\delta}\hat{\rho}(x)/|x|}\left(|\bar{x}|+c_{1}\delta_{0}+c_{2}\varepsilon_{0}\right),$

for some (new) constants $c_{1},c_{2}\geq 0$. We need to show that
$|R(x)|<\hat{\Delta}$ for all $\bar{x}$ and $x$ such that
$|x|,|\bar{x}|<\hat{\Delta}$. There are two cases to consider:

  * •

$|x|<\Delta+c_{0}<\hat{\Delta}$ for some $c_{0}>0$, and

  * •

$|x|\in[\Delta+c_{0},\hat{\Delta})$.

In the first case, solutions $\bar{x}$ (if they do exist) have to satisfy

$|\bar{x}|=|\widehat{H}^{x}(b,x,a,\varepsilon)|<\Delta+c_{0}+c_{1}\delta_{0}+c_{2}\varepsilon_{0}+\hat{\delta}C,$

where $C>0$ is is a constant in the interval
$(\sup_{|x|<\Delta+c_{0}}\rho(x),1)$. Hence

$|R(x)|\leq\Delta+c_{0}+2c_{1}\delta_{0}+2c_{2}\varepsilon_{0}+\hat{\delta}C<\hat{\Delta},$

is satisfied after choosing

$\hat{\delta}<\frac{\hat{\Delta}-(\Delta+c_{0})}{C}-2\frac{c_{1}\delta_{0}+c_{2}\varepsilon_{0}}{C}.$
(54)

Now assume that $|x|\in[\Delta+c_{0},\hat{\Delta})$. We use the fact that
$\rho(x)\geq C>0$ uniformly with respect to all such $x$. It follows that

$|R(x)|\leq\frac{1}{1+\hat{\delta}\hat{\rho}(x)/|x|}\left(\hat{\Delta}+c_{1}\delta_{0}+c_{2}\varepsilon_{0}\right)\leq\frac{1}{1+\hat{\delta}C/\hat{\Delta}}\left(\hat{\Delta}+c_{1}\delta_{0}+c_{2}\varepsilon_{0}\right)<\hat{\Delta}$

after choosing

$\hat{\delta}>\frac{c_{1}\delta_{0}+c_{2}\varepsilon_{0}}{C}.$ (55)

Combining (54) and (55) shows that $R:\hat{\mathcal{K}}\to\hat{\mathcal{K}}$
for all $\hat{\delta}$ such that

$\frac{c_{1}\delta_{0}+c_{2}\varepsilon_{0}}{C}<\hat{\delta}_{0}<\frac{\hat{\Delta}-(\Delta+c_{0})}{C}-2\frac{c_{1}\delta_{0}+c_{2}\varepsilon_{0}}{C},$

which, by choosing sufficiently small $\delta_{0},\varepsilon_{0}>0$, can
satisfied for any fixed

$0<\hat{\delta}<\frac{\hat{\Delta}-(\Delta+c_{0})}{C}.$ (56)

It remains to show that $R$ is a contraction. A direct calculation gives

$\begin{split}|R^{\prime}(x)|&\leq
A(x)^{-1}(\tilde{c}_{1}\delta_{0}+\tilde{c}_{2}\varepsilon_{0})+(A(x)^{-1})^{\prime}\left(|\bar{x}|+c_{1}\delta_{0}+c_{2}\varepsilon_{0}\right)\\\
&\leq\tilde{c}_{1}\delta_{0}+\tilde{c}_{2}\varepsilon_{0}+\left(|\bar{x}|+c_{2}\delta_{0}+c_{2}\varepsilon_{0}\right)\frac{\hat{\delta}|x\hat{\rho}^{\prime}(x)-\rho(x)|}{|x|^{2}},\end{split}$

where we used $A(x)\geq 1$. Notice that the right-hand-side is well-defined
for $x=0$, since $0\in\mathcal{K}$ and
$\hat{\rho}^{\prime}|_{\mathcal{K}}=\hat{\rho}|_{\mathcal{K}}\equiv 0$. In
particular,

$\frac{|x\hat{\rho}^{\prime}(x)-\rho(x)|}{|x|^{2}}\leq\frac{1}{\Delta^{2}}\sup_{|x|\in[\Delta,\hat{\Delta})}|x\hat{\rho}^{\prime}(x)-\rho(x)|\leq\frac{\hat{\Delta}M+1}{\Delta^{2}},$

where $M:=\sup_{x\in\hat{\mathcal{K}}}|x\hat{\rho}^{\prime}(x)|$. Hence

$|R^{\prime}(x)|\leq\tilde{c}_{1}\delta_{0}+\tilde{c}_{2}\varepsilon_{0}+\hat{\delta}\left(|\bar{x}|+c_{2}\delta_{0}+c_{2}\varepsilon_{0}\right)\left(\frac{\hat{\Delta}M+1}{\Delta^{2}}\right)<1,$

where the rightmost inequality is satisfied for sufficiently small
$\delta_{0},\varepsilon_{0}>0$ and

$\hat{\delta}<\frac{\Delta^{2}}{\hat{\Delta}(1+M\hat{\Delta})}.$ (57)

It follows that $R$ is a contraction as required. Finally, combining (56) and
(57) yields the bounds for $\hat{\delta}$ in (51), completing the proof of
statement (a).

The proof of statement (b) follows from arguments similar to those given in
the proof of (a) above. Specifically, $\widehat{H}_{R}$ is inflowing with
respect to $Y_{R}$ since

  * •

$(x_{R},y_{R})\in X_{R}\times Y_{R}\ \implies\
|\widehat{H}^{x}(x_{R},y_{R},\varepsilon)|<\hat{\Delta}$;

  * •

$(x_{R},y_{R})\in X_{R}\times Y_{R}\ \implies\
|\widehat{H}^{a}(x_{R},y_{R},\varepsilon)|<\delta_{0}$.

The former inequality was shown in the proof that
$R:\hat{\mathcal{K}}\to\hat{\mathcal{K}}$ above, and the latter follows from
(52). The fact that $\widehat{H}_{R}$ is outflowing with respect to $X_{R}$
follows immediately from (in fact, is the same as) the fact that by (53),
$\hat{b}=\widehat{H}^{b}(b,x,a,\varepsilon)$ has a unique solution with
$|\bar{b}|<\delta_{0}$. ∎

######  Remark 4.3.

Lemma 4.2 confirms the inflowing/outflowing hypotheses HMa)-b) in [54] for the
map (48), i.e. for the map (46). Note the crucial role played by the cutoff
function $\hat{\delta}\rho(x)n_{x}$. The fact that $\hat{\delta}$ must be
fixed and positive in accordance with (51), or more precisely, in accordance
with (55), shows that such cutoff procedures are necessary to satisfy the
hypotheses.

####  4.2.2 Conditions on relative contraction rates

We turn now to the verification of conditions on the relative contraction
rates. Such conditions are typically given in a quite general setting in terms
of Lipschitz-type bounds for the components of the map (48). Our aim here is
to restate these conditions in terms of spectral properties like normal
hyperbolicity which, as a spectral condition, is invariant under
$C^{1}-$smooth coordinate transformations and thus detectable in the original
coordinates of (36). The cost of this reformulation is that we require at
least $C^{1}-$smoothness of the map, while only Lipschitz continuity is
required in [54]. A similar price is paid in Fenichel theory [17, 73].

Let $i\in\\{A,R\\}$ in order to streamline sub/superscript notations wherever
possible. Since the maps $\widehat{H}_{i}$ are $C^{1}-$smooth in
$(x_{i},y_{i},\varepsilon)$, there exist constants
$\Gamma^{i}_{11},L^{i}_{12},L^{i}_{13},L^{i}_{21},L^{i}_{22},L^{i}_{23}\geq 0$
such that the following Lipschitz-type estimates are satisfied for any fixed
choice of $x_{i,1},x_{i,2}\in X_{i}$, $y_{i,1},y_{i,2}\in Y_{i}$ and
$\varepsilon_{1},\varepsilon_{2}\in[0,\varepsilon_{0})$:

$\begin{split}|\widehat{H}^{x_{i}}_{i}(x_{i,1},y_{i},\varepsilon)-\widehat{H}^{x_{i}}_{i}(x_{i,2},y_{i},\varepsilon)|&\geq\Gamma^{i}_{11}|x_{i,1}-x_{i,2}|,\\\
|\widehat{H}^{x_{i}}_{i}(x_{i},y_{i,1},\varepsilon_{1})-\widehat{H}^{x_{i}}_{i}(x_{i},y_{i,2},\varepsilon_{2})|&\leq
L^{i}_{12}|y_{i,1}-y_{i,2}|+L^{i}_{13}|\varepsilon_{1}-\varepsilon_{2}|,\\\
|\widehat{H}^{y_{i}}_{i}(x_{i,1},y_{i,1},\varepsilon_{1})-\widehat{H}^{y_{i}}_{i}(x_{i,2},y_{i,2},\varepsilon_{2})|&\leq
L^{i}_{21}|x_{i,1}-x_{i,2}|+L^{i}_{22}|y_{i,1}-y_{i,2}|+L^{i}_{23}|\varepsilon_{1}-\varepsilon_{2}|.\end{split}$
(58)

Smoothness allows for a characterisation of these Lipschitz-type bounds in
terms of partial derivatives.

######  Lemma 4.4.

The following choices satisfy the conditions in (58) for the map
$\widehat{H}_{A}$:

$\displaystyle\Gamma_{11}^{A}$
$\displaystyle=1+r^{A}_{11}(\delta_{0},\varepsilon_{0}),$ (59) $\displaystyle
L_{12}^{A}$
$\displaystyle=\sup_{x\in\hat{\mathcal{K}}}\left\lVert(\tilde{N}^{x}N^{y}_{a})|_{\hat{S}}\right\rVert+r_{12}^{A}(\delta_{0},\varepsilon_{0}),$
$\displaystyle L_{13}^{A}$
$\displaystyle=\sup_{x\in\hat{\mathcal{K}}}\sqrt{|\tilde{G}^{x}|_{\hat{S}}|^{2}+|((N_{r}^{y})^{-1}DfG)|_{\hat{S}}|^{2}}+r_{13}^{A}(\delta_{0},\varepsilon_{0}),$
$\displaystyle L_{21}^{A}$
$\displaystyle=r_{21}^{A}(\delta_{0},\varepsilon_{0}),$ $\displaystyle
L_{22}^{A}$ $\displaystyle=\nu_{A}+r_{22}^{A}(\delta_{0},\varepsilon_{0}),$
$\displaystyle L_{23}^{A}$
$\displaystyle=\sup_{x\in\hat{\mathcal{K}}}|((N^{y}_{a})^{-1}DfG)|_{\hat{S}}|+r_{23}^{A}(\varepsilon_{0}),$

where the functions $r_{kj}^{A}(\delta_{0},\varepsilon_{0})$ and
$r_{23}^{A}(\varepsilon_{0})$ are continuous such that
$r_{kj}^{A}(0,0)=r_{23}^{A}(0)=0$.

The following choices satisfy the conditions in (58) for the map
$\widehat{H}_{R}$:

$\displaystyle\Gamma_{11}^{R}$
$\displaystyle=\nu_{R}+r_{11}^{R}(\delta_{0},\varepsilon_{0}),$ (60)
$\displaystyle L_{12}^{R}$
$\displaystyle=r_{12}^{R}(\delta_{0},\varepsilon_{0}),$ $\displaystyle
L_{13}^{R}$
$\displaystyle=\sup_{x\in\hat{\mathcal{K}}}|((N^{y}_{r})^{-1}DfG)|_{\hat{S}}|+r_{13}^{R}(\varepsilon_{0}),$
$\displaystyle L_{21}^{R}$
$\displaystyle=\sup_{x\in\hat{\mathcal{K}}}\left\lVert(\tilde{N}^{x}N^{y}_{r})|_{\hat{S}}\right\rVert+r_{21}^{R}(\delta_{0},\varepsilon_{0}),$
$\displaystyle L_{22}^{R}$
$\displaystyle=1+\hat{\delta}+r_{22}^{R}(\delta_{0},\varepsilon_{0}),$
$\displaystyle L_{23}^{R}$
$\displaystyle=\sup_{x\in\hat{\mathcal{K}}}\sqrt{|\tilde{G}^{x}|_{\hat{S}}|^{2}+|((N_{a}^{y})^{-1}DfG)|_{\hat{S}}|^{2}}+r_{23}^{R}(\delta_{0},\varepsilon_{0}),$

where the functions $r^{R}_{kj}(\delta_{0},\varepsilon_{0})$ and
$r^{R}_{13}(\varepsilon_{0})$ are continuous such that
$r_{kj}^{R}(0,0)=r_{13}^{R}(0)=0$.

###### Proof.

Since both $\widehat{H}_{i}$ are at least $C^{1}-$smooth in all arguments,
partial derivatives are locally bounded and the following choices for
$\Gamma^{i}_{11},L^{i}_{12},L^{i}_{13},L^{i}_{21},L^{i}_{22}$, $L^{i}_{23}\geq
0$ satisfy (58):

$\begin{split}\Gamma_{11}^{i}=\inf_{({x_{i}},{y_{i}},\varepsilon)\in
U_{i}}\left\lVert
D_{x_{i}}\widehat{H}^{x_{i}}_{i}({x_{i}},{y_{i}},\varepsilon)\right\rVert,\qquad&L_{21}^{i}=\sup_{({x_{i}},{y_{i}},\varepsilon)\in
U_{i}}\left\lVert
D_{x_{i}}\widehat{H}^{y_{i}}_{i}({x_{i}},{y_{i}},\varepsilon)\right\rVert,\\\
L_{12}^{i}=\sup_{({x_{i}},{y_{i}},\varepsilon)\in U_{i}}\left\lVert
D_{y_{i}}\widehat{H}^{x_{i}}_{i}({x_{i}},{y_{i}},\varepsilon)\right\rVert,\qquad&L_{22}^{i}=\sup_{({x_{i}},{y_{i}},\varepsilon)\in
U_{i}}\left\lVert
D_{y_{i}}\widehat{H}^{y_{i}}_{i}({x_{i}},{y_{i}},\varepsilon)\right\rVert,\\\
L_{13}^{i}=\sup_{({x_{i}},{y_{i}},\varepsilon)\in U_{i}}\left\lVert
D_{\varepsilon}\widehat{H}^{x_{i}}_{i}({x_{i}},{y_{i}},\varepsilon)\right\rVert,\qquad&L_{23}^{i}=\sup_{({x_{i}},{y_{i}},\varepsilon)\in
U_{i}}\left\lVert
D_{\varepsilon}\widehat{H}^{y_{i}}_{i}({x_{i}},{y_{i}},\varepsilon)\right\rVert,\end{split}$
(61)

where $U_{i}:=X_{i}\times Y_{i}\times[0,\varepsilon_{0})$. We omit the
subsequent derivations for all expressions except $\Gamma_{11}^{A}$ and
$L_{22}^{R}$, since these can be derived by a direct evaluation of the
corresponding expression in (61), followed (if necessary) by the application
of some standard triangle-type inequalities.

To obtain the expression for $\Gamma_{11}^{A}$, notice that Taylor expanding
about $\hat{S}:a=b=\varepsilon=0$ gives

$\begin{split}\Gamma_{11}^{A}&=\inf_{(x_{A},y_{A},\varepsilon)\in
U_{A}}\left\lVert\begin{pmatrix}D_{b}\widehat{H}^{b}(b,x,a,\varepsilon)&D_{x}\widehat{H}^{b}(b,x,a,\varepsilon)\\\
D_{b}\widehat{H}^{x}(b,x,a,\varepsilon)&D_{x}\widehat{H}^{x}(b,x,a,\varepsilon)\end{pmatrix}\right\rVert\\\
&=\inf_{x\in\hat{\mathcal{K}}}\left\lVert\begin{pmatrix}D_{b}\widehat{H}^{b}&D_{x}\widehat{H}^{b}\\\
D_{b}\widehat{H}^{x}&D_{x}\widehat{H}^{x}\end{pmatrix}\bigg{|}_{\hat{S}}\right\rVert+r_{11}^{A}(\delta_{0},\varepsilon_{0})\\\
&=\inf_{x\in\hat{\mathcal{K}}}\left\lVert\begin{pmatrix}I_{n_{r}}+\Lambda_{r}&O_{n_{r}\times
k}\\\
\tilde{N}^{x}N^{y}_{r}&I_{k}+\hat{\delta}D_{x}(\rho(x)n_{x})\end{pmatrix}\bigg{|}_{\hat{S}}\right\rVert+r_{11}^{A}(\delta_{0},\varepsilon_{0}),\end{split}$

where $r_{11}^{A}(\delta_{0},\varepsilon_{0})$ is continuous and satisfies
$r_{11}^{A}(0,0)=0$. Since $\left\lVert I_{k}\right\rVert=1$, the matrix norm
in the last line must be greater than or equal to $1$. Hence
$\Gamma_{11}^{A}\geq 1+r_{11}^{A}(\delta_{0},\varepsilon_{0})$, thereby
justifying the choice for $\Gamma_{11}^{A}$ in (59).

Now consider $L_{22}^{R}$. Similar calculations lead to

$L_{22}^{R}=\sup_{x\in\hat{\mathcal{K}}}\left\lVert\begin{pmatrix}I_{k}+\hat{\delta}D_{x}(\rho(x)n_{x})&(\tilde{N}^{x}N^{y}_{a})(x,0,0)\\\
O_{n_{a}\times
k}&I_{n_{a}}+\Lambda_{a}(x,0,0)\end{pmatrix}\right\rVert+r_{22}^{R}(\delta_{0},\varepsilon_{0})=:\left\lVert
B\right\rVert+\tilde{r}_{22}^{R}(\delta_{0},\varepsilon_{0}),$

for a continuous function $\tilde{r}_{22}^{R}(\delta_{0},\varepsilon_{0})$
such that $\tilde{r}_{22}^{R}(0,0)=0$. We need a sufficiently sharp estimate
for $\left\lVert B\right\rVert$ which, by construction, has an associated
operator norm property

$\left\lVert B\begin{pmatrix}x\\\ b\end{pmatrix}\right\rVert\leq
L^{R}_{22}\left\lvert\begin{pmatrix}x\\\ b\end{pmatrix}\right\rvert$ (62)

for sufficiently small but fixed $\delta_{0},\varepsilon_{0}>0$, uniformly
with respect to $(b,x)\in Y_{R}$. Direct estimates yield

$\begin{split}\left\lVert B\begin{pmatrix}x\\\
b\end{pmatrix}\right\rVert^{2}&=\left\lvert\begin{pmatrix}(I_{k}+\hat{\delta}D_{x}(\rho(x)n_{x}))x+(\tilde{N}^{x}N^{y}_{a})(x,0,0)b\\\
(I_{n_{a}}+\Lambda_{a}(x,0,0))b\end{pmatrix}\right\rvert^{2}\\\
&\leq(1+\hat{\delta})^{2}\hat{\Delta}^{2}+c_{1}\hat{\Delta}\delta_{0}+c_{2}\delta_{0}^{2}+\nu_{A}^{2}\delta_{0}^{2}\\\
&\leq\tilde{C}^{2}\left\lvert\begin{pmatrix}x\\\
b\end{pmatrix}\right\rvert^{2}\\\
&\leq\tilde{C}^{2}(\hat{\Delta}^{2}+\delta_{0}^{2}),\end{split}$

where $c_{1},c_{2}\geq 0$ are constants and we require that

$\tilde{C}\geq\sqrt{\frac{(1+\hat{\delta})^{2}\hat{\Delta}^{2}+c_{1}\hat{\Delta}\delta_{0}+c_{2}\delta_{0}^{2}+\nu_{A}^{2}\delta_{0}^{2}}{\hat{\Delta}^{2}+\delta_{0}^{2}}}=1+\hat{\delta}+\hat{r}_{22}^{R}(\delta_{0}),$

where the function $\hat{r}_{22}^{R}(\delta_{0})$ is continuous with
$\hat{r}_{22}^{R}(0)=0$. Choosing the minimal such $\tilde{C}$ and combining
with (62) leads to the expression for $L_{22}^{R}$ in (60). ∎

We now use the expressions in Lemma 4.4 in order to check the relevant
contractivity/repulsivity conditions. In order to simplify notation in the
following we define

$\omega_{i}:=\frac{2L^{i}_{12}L^{i}_{21}}{\Gamma^{i}_{11}-L^{i}_{22}+\sqrt{(\Gamma^{i}_{11}-L^{i}_{22})^{2}-4L^{i}_{12}L^{i}_{21}}}$

for both $i\in\\{A,R\\}$.

######  Lemma 4.5.

(c.f. [54] conditions CM, CMA and CMR) For sufficiently small
$\delta_{0},\varepsilon_{0}>0$ and $\hat{\delta}$ such that

$\nu_{R}>1+\hat{\delta},$ (63)

the following conditions are satisfied:

  * (i)

In both cases $i\in\\{A,R\\}$, we have

$2\sqrt{L^{i}_{12}L^{i}_{21}}<\Gamma^{i}_{11}-L_{22}^{i}.$

  * (ii)

In case $i=A$ we have

$L_{22}^{A}+\omega_{A}<1.$

  * (iii)

In case $i=R$ we have

$\Gamma^{R}_{11}-\omega_{R}>1.$

###### Proof.

This is immediate given the expressions in Lemma 4.4. Since
$L^{i}_{12}L^{i}_{21}\to 0$ as $(\delta_{0},\varepsilon_{0})\to(0,0)$, the
condition (i) can always be satisfied as long as
$0<\Gamma^{i}_{11}-L_{22}^{i}$. The latter conditions can be checked directly.
One the one hand we have

$\Gamma^{A}_{11}-L_{22}^{A}=1-\nu_{A}-(r_{11}^{A}(\delta_{0},\varepsilon_{0})+r_{22}^{A}(\delta_{0},\varepsilon_{0}))>0,$

for sufficiently small $\delta_{0},\varepsilon_{0}>0$ since $\nu_{A}<1$. On
the other hand we have

$\Gamma^{R}_{11}-L_{22}^{R}=\nu_{R}-(1+\hat{\delta})-(r_{11}^{R}(\delta_{0},\varepsilon_{0})+r_{22}^{R}(\delta_{0},\varepsilon_{0}))>0,$

for sufficiently small $\delta_{0},\varepsilon_{0}>0$ and $\hat{\delta}$ in
the interval (63). In order to verify (ii)-(iii), observe that $\omega_{i}\to
0$ as $(\delta_{0},\varepsilon_{0})\to(0,0)$ in both cases $i\in\\{A,R\\}$.
The results in (ii) and (iii) follow immediately since $L_{22}^{A}<1$ and
$\Gamma^{R}_{11}>1$ for sufficiently small $\delta_{0},\varepsilon_{0}>0$. ∎

####  4.2.3 Smoothness conditions

It remains to check a number of conditions relating to smoothness of the
invariant slow manifolds and foliations. We start with conditions for the
smoothness of slow manifolds.

######  Lemma 4.6.

(c.f. [54] conditions CMR(k) and CMA(k) for smoothness of slow manifolds)
Consider the $C^{r\geq 1}$-smooth map (48). Then for sufficiently small
$\delta_{0},\varepsilon_{0}>0$, and $\hat{\delta}$ such that

$\nu_{R}>(1+\hat{\delta})^{r}>1,$ (64)

the following conditions are satisfied:

  * (i)

In case $i=A$ we have

$L_{22}^{A}+\omega_{A}<(\Gamma_{11}^{A}-\omega_{A})^{r}.$

  * (ii)

In case $i=R$ we have

$(L_{22}^{R}+\omega_{R})^{r}<\Gamma^{R}_{11}-\omega_{R}.$

###### Proof.

Straightforward calculations using the expressions in Lemma 4.4. The
requirement (64) is necessary in (ii). ∎

Finally we need to verify a number of conditions relating to smoothness of the
invariant foliations corresponding to persisting stable and unstable
manifolds. These can be checked directly using the expressions in Lemma 4.4 as
well as the quantities

$L_{11}^{A}:=\sup_{({x_{A}},{y_{A}},\varepsilon)\in U_{A}}\left\lVert
D_{x_{A}}\widehat{H}^{x_{A}}_{A}({x_{A}},{y_{A}},\varepsilon)\right\rVert,\qquad\Gamma_{22}^{i}:=\inf_{({x_{i}},{y_{i}},\varepsilon)\in
U_{i}}\left\lVert
D_{y_{i}}\widehat{H}^{y_{i}}_{i}({x_{i}},{y_{i}},\varepsilon)\right\rVert.$

For $L_{11}^{A}$ and $\Gamma_{22}^{R}$, arguments based on direct estimates
and operator norm properties similar to those applied for $\Gamma_{11}^{A}$ in
the proof of Lemma 4.4 lead to concrete (though potentially less than optimal)
expressions

$L_{11}^{A}=1+\hat{\delta}+\tilde{r}_{11}^{A}(\delta_{0},\varepsilon_{0}),\qquad\Gamma_{22}^{R}=1+\tilde{r}_{22}^{R}(\delta_{0},\varepsilon_{0}),$
(65)

where $\tilde{r}^{A}_{11}(\delta_{0},\varepsilon_{0})$ and
$\tilde{r}^{R}_{22}(\delta_{0},\varepsilon_{0})$ are continuous such that
$\tilde{r}^{A}_{11}(0,0)=\tilde{r}^{R}_{22}(0,0)=0$. For $\Gamma_{22}^{R}$,
direct calculations yield

$\Gamma_{22}^{A}=\inf_{x\in\hat{\mathcal{K}}}\left\lVert
I_{n_{a}}+\Lambda_{a}|_{\hat{S}}\right\rVert+\tilde{r}^{A}_{22}(\delta_{0},\varepsilon_{0}),$
(66)

where $\tilde{r}^{A}_{22}(\delta_{0},\varepsilon_{0})$ is continuous such that
$\tilde{r}^{A}_{22}(0,0)=0$. It is important to note that $\Gamma_{22}^{A}=0$
is possible under superstable conditions in which all $n_{a}$ multipliers of
$I_{n_{a}}+\Lambda_{a}|_{\hat{S}}$ have real part zero.

######  Lemma 4.7.

(c.f. [54] conditions CMB, CMAB($k-1$) and CMRB($k-1$) for smoothness of the
foliations) Consider the $C^{r\geq 1}$-smooth map (48), and assume that
$I_{n_{a}}+\Lambda_{a}|_{\hat{S}}$ has at least one multiplier $\mu_{a,j}(x)$
such that

$|\mu_{a,j}(x)|>\tilde{c}>0$ (67)

for some $x\in\hat{\mathcal{K}}$, for some constant $\tilde{c}>0$. Then for
sufficiently small $\delta_{0},\varepsilon_{0}>0$, and $\hat{\delta}$ such
that

$\nu_{A}<\frac{1}{(1+\hat{\delta})^{r-1}}<1+\hat{\delta}<\nu_{R},$ (68)

the following conditions are satisfied:

  * (i)

In both cases $i\in\\{A,R\\}$ we have

$\Gamma_{22}^{i}-\omega_{i}>0.$

  * (ii)

In case $i=A$ we have

$\frac{\Gamma^{A}_{11}-\omega_{A}}{L_{22}^{A}+\omega_{A}}>(L_{11}^{A}+\omega_{A})^{r-1}.$

  * (iii)

In case $i=R$ we have

$\frac{L^{R}_{22}+\omega_{R}}{\Gamma_{11}^{R}-\omega_{R}}<(\Gamma_{22}^{R}-\omega_{R})^{r-1}.$

###### Proof.

The condition (i) is immediate in case $i=R$. In case $i=A$, it follows from
the requirement $|\mu_{a,j}(x)|>\tilde{c}>0$ for some
$j\in\\{1,\ldots,n_{a}\\}$ and $x\in\hat{\mathcal{K}}$ since in this case,

$\left\lVert
I_{n_{a}}+\Lambda_{a}|_{\hat{S}}\right\rVert\geq|\mu_{a,j}(x)|>\tilde{c}>0.$

Conditions (ii)-(iii) are verified directly using equations (65)-(66) and the
expressions in Lemma 4.4. Specifically, the inequality in (ii) follows from
the fact that

$\frac{\Gamma_{11}^{A}}{L_{22}^{A}}\sim\frac{1}{\nu_{A}}>(1+\hat{\delta})^{r-1}$

under (68) as $(\delta_{0},\varepsilon_{0})\to(0,0)$, and the inequality in
(iii) follows from the fact that

$\frac{L_{22}^{R}}{\Gamma_{11}^{R}}\sim\frac{1+\hat{\delta}}{\nu_{R}}<1$

as $(\delta_{0},\varepsilon_{0})\to(0,0)$. ∎

######  Remark 4.8.

The local condition on the stable multipliers in (67) rules out the
possibility of degenerate local superstability in the sense that we disallow
the case where $\mu_{a,j}(x)\equiv 0$ for all $x\in\hat{\mathcal{K}}$ and
$j=1,\ldots,n_{a}$.

###  4.3 Proof of the main results

We are now in a position to prove the main results in Section 3. Since it
allows for a simpler proof of the slow manifold Theorems 3.1 and 3.2, we first
prove persistence of the stable and unstable manifolds $W^{s/u}_{loc}(S)$
defined in (19) for $0<\varepsilon\ll 1$.

We shall consider the map (48) throughout, for any $\hat{\delta}$ fixed in the
interval

$0<\hat{\delta}<\min\left\\{\nu_{R}^{1/r}-1,\frac{\hat{\Delta}-(\Delta+c_{0})}{C},\frac{\Delta^{2}}{\hat{\Delta}(1+M\hat{\Delta})}\right\\}.$
(69)

This ensures that the results of Section 4.2 apply for
$\varepsilon_{0},\delta_{0}>0$ sufficiently small.

####  4.3.1 Proof of Theorems 3.3

We first prove persistence of the local stable manifold $W^{s}_{loc}(S)$. It
follows from Lemmas 4.2, 4.4 and 4.5 that Theorem [54, Theorem 1.3] applies to
the map (48) in the repelling formulation $\widehat{H}_{R}$, see again (50).
This yields the existence of a positively invariant manifold

$\mathcal{M}^{s}:=\left\\{(\varphi_{R}(y_{R},\varepsilon),y_{R}):y_{R}\in
Y_{R}\right\\}\subseteq X_{R}\times Y_{R},$

where the function $\varphi_{R}(y_{R},\varepsilon)$ is (at least) uniformly
Lipschitz continuous and satisfies the invariance equation

$\bar{x}_{R}=\widehat{H}_{R}^{x_{R}}\left(\varphi_{R}(y_{R},\varepsilon),y_{R},\varepsilon\right)=\varphi_{R}\left(\widehat{H}_{R}^{y_{R}}\left(\varphi_{R}(y_{R},\varepsilon),y_{R},\varepsilon\right),\varepsilon\right),$
(70)

for all $y_{R}=(x,a)^{\textnormal{T}}$ such that
$\widehat{H}_{R}^{y_{R}}(\varphi_{R}(y_{R},\varepsilon),y_{R},\varepsilon)\in
Y_{R}$. Combining this with Lemma 4.5, it follows after an application of [54,
Theorem 3.1] that the function $\varphi_{R}$, and hence the manifold
$\mathcal{M}^{s}$, is $C^{r}-$smooth in both $y_{R}$ and $\varepsilon$.

An analogous application of the preceding arguments in the case that
$\varepsilon=0$ leads to

$\mathcal{M}^{s}|_{\varepsilon=0}=W^{s}_{loc}(\hat{S}),$

where $W^{s}_{loc}(\hat{S})$ denotes the smooth extension of the stable
manifold $W^{s}_{loc}(S)$ defined in (19) onto the enlarged domain
$X_{A}\times Y_{A}$ with $x\in\hat{\mathcal{K}}$. From this observation, one
may conclude that $\mathcal{M}^{s}$ and $W^{s}_{loc}(\hat{S})$ are
$O(\varepsilon)-$close for $\varepsilon\in(0,\varepsilon_{0})$ by a direct
application of [54, Theorem 2.4]. Finally, a locally (as opposed to globally)
invariant manifold is obtained by restricting $\mathcal{M}^{s}$ to the
original domain with $x\in\mathcal{K}\subset\hat{\mathcal{K}}$. This yields a
new manifold given in $(b,x,a)-$coordinates as a graph

$\mathcal{M}^{s}|_{B\times\mathcal{K}\times
A}=:W^{s}_{loc}(S_{\varepsilon})=\left\\{(\varphi_{R}(x,a,\varepsilon),x,a):(x,a)\in\mathcal{K}\times
A\right\\},$ (71)

where in particular we have $\varphi_{R}(x,a,\varepsilon)=O(\varepsilon)$,
since $W^{s}_{loc}(S)$ and $W^{s}_{loc}(S_{\varepsilon})$ are
$O(\varepsilon)-$close and
$W^{s}_{loc}(S)=\\{(0,x,a):(x,a)\in\mathcal{K}\times A\\}$ for the map
(48)$|_{\varepsilon=0}$. The local invariance property of
$W^{s}_{loc}(S_{\varepsilon})$ in Theorem 3.3 follows from the global
(positive) invariance of $\mathcal{M}^{s}$, i.e. the property that
$\widehat{H}_{R}(\mathcal{M}^{s},\varepsilon)\subseteq\mathcal{M}^{s}$, after
restricting to $B\times\mathcal{K}\times A$. The preceding arguments show that
the perturbed stable manifold $W^{s}_{loc}(S_{\varepsilon})$ has a graph
representation which locally satisfies the relevant properties (i), (ii) and
(iii) of Theorem 3.3.

Persistence of the local unstable manifold $W^{u}_{loc}(S)$ follows using
similar arguments to those presented for the persistence of $W^{s}_{loc}(S)$,
so we shall restrict ourselves to an overview of the proof. Applying [54,
Theorem 1.5] to (48) in the attracting formulation $\widehat{H}_{A}$ in (49)
yields the existence of a negatively invariant manifold

$\mathcal{M}^{u}:=\left\\{\left(x_{A},\varphi_{A}(x_{A},\varepsilon)\right):x_{A}\in
X_{A}\right\\}\subseteq X_{A}\times Y_{A},$

where the function $\varphi_{A}(x_{A},\varepsilon)$ is $C^{r}-$smooth by [54,
Theorem 3.6] and satisfies the invariance equation

$\bar{y}_{A}=\widehat{H}_{A}^{y_{A}}\left(x_{A},\varphi_{A}(x_{A},\varepsilon),\varepsilon\right)=\varphi_{A}\left(\widehat{H}_{A}^{x_{A}}\left(x_{A},\varphi_{A}(x_{A},\varepsilon),\varepsilon\right),\varepsilon\right).$

Consideration of the $\varepsilon=0$ case shows that
$\mathcal{M}^{u}|_{\varepsilon=0}=W^{u}_{loc}(\hat{S})$, which is
$O(\varepsilon)-$close to $\mathcal{M}^{u}$ when
$\varepsilon\in(0,\varepsilon_{0})$ due to [54, Theorem 2.1]. Restricting to
$\mathcal{K}\subset\hat{\mathcal{K}}$ yields the (locally invariant) perturbed
unstable manifold

$W^{u}_{loc}(S_{\varepsilon}):=\left\\{(b,x,\varphi_{A}(b,x,\varepsilon)):(b,x)\in
B\times\mathcal{K}\right\\},$

described in Theorem 3.3, where $\varphi_{A}(b,x,\varepsilon)=O(\varepsilon)$.
Note that we also gain invertibility of the restricted map
$\widehat{H}_{A}|_{\mathcal{M}^{u}}$ via [54, Theorem 1.5], implying
invertibility of $H|_{W^{u}_{loc}(S_{\varepsilon})}$ as required in Theorem
3.3. This allows for a straightforward derivation of the negative invariance
condition in assertion (iii) using the fact that
$\mathcal{M}^{u}\subseteq\widehat{H}_{A}(\mathcal{M}^{u},\varepsilon)$.

Aside from the final assertion about the intersection
$S_{\varepsilon}=W^{s}_{loc}(S_{\varepsilon})\cap
W^{u}_{loc}(S_{\varepsilon})$, which will be considered in the next section,
the preceding arguments combine to prove a local graph formulation of Theorem
3.3. Standard arguments using compactness and a partition of unity complete
the proof. ∎

####  4.3.2 Proof of Theorems 3.1 and 3.2

We now prove Theorem 3.2, which is sufficient to prove Theorem 3.1.

Much of the content of Theorem 3.2 can be derived using Theorem 3.3 and its
proof in Section 4.3.1 by defining the ‘slow manifold’ $S_{\varepsilon}$ as
the intersection of perturbed stable and unstable manifolds
$W^{s}_{loc}(S_{\varepsilon})$ and $W^{u}_{loc}(S_{\varepsilon})$. For the map
(48), the proof of Theorem 3.3 in in Section 4.3.1 together with [54, Theorem
1.7] implies that this intersection is well-defined and given by

$S_{\varepsilon}:=W^{s}_{loc}(S_{\varepsilon})\cap
W^{u}_{loc}(S_{\varepsilon})=\left\\{(r_{b}(x),x,r_{a}(x)):x\in\mathcal{K}\right\\},$
(72)

where $r_{b}(x)$ and $r_{a}(x)$ are $C^{r}$-smooth functions which satisfy

$r_{b}(x)=O(\varepsilon),\qquad r_{a}(x)=O(\varepsilon).$ (73)

Invertibility of the restricted map $H|_{S_{\varepsilon}}$ follows from the
fact that $H|_{\mathcal{M}^{u}}$ is invertible, since
$S_{\varepsilon}\subset\hat{S}_{\varepsilon}\subseteq\mathcal{M}^{u}$. The
fact that $S_{\varepsilon}$ is $O(\varepsilon)-$close and diffeomorphic to the
critical manifold $S=\\{(x,0):x\in\mathcal{K}\\}$ follows by (72) and (73).
The local invariance of $S_{\varepsilon}$ described in Theorem 3.2 follows
from the global invariance
$H(\hat{S}_{\varepsilon},\varepsilon)=\hat{S}_{\varepsilon}$ implied by [54,
Theorem 1.7] after restriction to $\mathcal{K}\subset\widehat{\mathcal{K}}$,
and the particular form of the invariance equations (39) and (40) in Theorem
3.2 follow after expressing the local invariance requirement

$z_{\varepsilon}\in S_{\varepsilon},\
H(z_{\varepsilon},\varepsilon)\in\mathcal{U}\quad\implies\quad\bar{z}_{\varepsilon}=H(z_{\varepsilon},\varepsilon)\in
S_{\varepsilon},$

in the local $(x,y)$ coordinates of (37). Finally, the form of the expansion
(38) follows from its derivation in the proof of Proposition 2.14, see again
equation (23) and Remark 2.15.

Thus we have proved Theorem 3.2 and, correspondingly, the local expression of
Theorem 3.1. Extending the local result via compactness and a partition of
unity completes the proof of Theorem 3.1. ∎

####  4.3.3 Proof of Theorem 3.4

Applying [54, Theorem 4.1] to the map (48) in the repelling formulation
$\widehat{H}_{R}$, we find that the perturbed stable manifold
$W^{s}_{loc}(\hat{S}_{\varepsilon})$ of the map (48) admits a foliation by
stable fibers $w^{s}_{loc}(z_{\varepsilon})$ with base points
$z_{\varepsilon}\in\hat{S}_{\varepsilon}$, which are $C^{r}-$smooth in
$\varepsilon$ and satisfy the (global) positive invariance property

$\widehat{H}_{R}(w^{s}_{loc}(z_{\varepsilon}))\subseteq
w^{s}_{loc}\left(\widehat{H}_{R}(z_{\varepsilon})\right),$

for all $z_{\varepsilon}\in\hat{S}_{\varepsilon}$. This motivates the
definition

$W^{s}_{loc}(\hat{S}_{\varepsilon}):=\bigcup_{z_{\varepsilon}\in\hat{S}_{\varepsilon}}w_{loc}^{s}(z_{\varepsilon}).$
(74)

Theorem 4.1 in [54] also ensures that the fibers
$w_{loc}^{s}(z_{\varepsilon})$ are continuous and identical to the set of
points $z_{R}\in X_{R}\times Y_{R}$ for which iterates
$\widehat{H}_{R}^{j}(z_{R},\varepsilon)$ are exponentially attracted to
iterates $\widehat{H}_{R}^{j}(z_{\varepsilon},\varepsilon)$ along
$\hat{S}_{\varepsilon}$. More precisely, $w_{loc}^{s}(z_{\varepsilon})$ is the
set of points $z_{R}\in X_{R}\times Y_{R}$ such that for each $j\in\mathbb{N}$
we have

$\big{|}\widehat{H}_{R}^{j}(z_{R},\varepsilon)-\widehat{H}_{R}^{j}(z_{\varepsilon},\varepsilon)\big{|}\leq
c\chi_{A}^{j},$

for some $c>0$ and any fixed $\chi_{A}\in(\nu_{A},1)$. Since the limit
$\varepsilon\to 0$ is defined (fibers are $C^{r}-$smooth in $\varepsilon$) and
the stable fibers for the layer map also admit such a characterisation, it
follows that $W_{loc}^{s}(\hat{S}_{\varepsilon})$ converges fiberwise to
$W_{loc}^{s}(\hat{S})$ as $\varepsilon\to 0$. The fact that the fibers are
$C^{r-1}-$smooth overall, i.e. with respect to coordinate and base point
variation, follows from [54, Theorem 5.1]. The nondegeneracy condition in
assertion (v) is a consequence of the requirement (67) imposed in order to
prove Lemma 4.7, see also Remark 4.8. Smoothness and convergence for
$\varepsilon\to 0$ together imply that $W_{loc}^{s}(\hat{S}_{\varepsilon})$
and $W_{loc}^{s}(\hat{S})$ are $O(\varepsilon)-$close. Restricting to the
original domain $\mathcal{K}\subset\hat{\mathcal{K}}$ yields the desired
results. In particular, restricting to $\mathcal{K}$ amounts to a restriction
of (74) to $S_{\varepsilon}\subset\hat{S}_{\varepsilon}$, yielding the stable
foliation $W^{s}_{loc}(S_{\varepsilon})\subset
W^{s}_{loc}(\hat{S}_{\varepsilon})$ from equation (42). The local invariance
and contraction properties in assertions (i) and (iii) follow from the
corresponding global invariance and contraction properties identified above on
the enlarged domain.

Similar arguments based on [54, Theorems 4.2 and 5.2] prove the corresponding
results for the unstable foliation; here we omit the details for brevity,
noting only that the additional requirement that $H$ is invertible for along
unstable fibers $w^{u}_{loc}(z_{\varepsilon})$ follows from the invertibility
of $H$ on $W^{u}_{loc}(S_{\varepsilon})$ and equation (42), see again Theorem
3.3 (iii). The preceding arguments prove Theorem 3.4 locally, and can be
extended over compact domains in order to complete the proof using a partition
of unity. ∎

##  5 Applications

Having proved the main results, we consider a number of applications of the
DGSPT formalism developed in Sections 2-3. We begin in Section 5.1 with a
geometric analysis of a two-dimensional map-based model for neuronal dynamics
originally presented in [5]. This example will help to demonstrate the
application of the theory, and is intended to provide a kind of ‘benchmark