Symmetry-breaking singular controller design for Bogdanov-Takens bifurcations
with an application to Chua system

Majid Gazora,b ${}^{\dagger}\,$Corresponding author. Phone: (98-31) 33913634;
Fax: (98-31) 33912602; Email<EMAIL_ADDRESS>Email<EMAIL_ADDRESS>and
Nasrin Sadrib,†

aDepartment of Mathematical Sciences, Isfahan University of Technology
Isfahan 84156-83111, Iran
bSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box: 19395-5746, Tehran, Iran

###### Abstract

We provide a complete symmetry-breaking bifurcation control for
$\mathbb{Z}_{2}$-equivariant smooth differential systems with Bogdanov-Takens
singularities. Controller coefficient space is partitioned by critical
controller sets into different connected regions. The connected regions
provide a classification for all qualitatively different dynamics of the
controlled system. Hence, a state feedback controller design with four small
controller coefficients is proposed for an efficient and full singular
symmetry-breaking control. We show that our approach works well for nonlinear
control systems with both controllable and uncontrollable linearizations.
Asymmetric bifurcations are all associated with the controlled system and they
start with a primary controlled pitchfork bifurcation from the origin. Origin
is a primary equilibrium for the uncontrolled system. This gives rise to two
secondary local equilibria $E_{\pm}$ for the controlled system. These
equilibria further experience tertiary fold and hysteresis type bifurcations.
The secondary and primary equilibria experience Hopf and Bautin bifurcations
leading to the appearance of one limit cycle $\mathscr{C}_{0}$ from primary
equilibrium, and either one or two from each secondary equilibria; namely,
$\mathscr{C}^{1}_{\pm}$ and $\mathscr{C}^{2}_{\pm}$. The collisions of limit
cycles ($\mathscr{C}_{0}$ and $\mathscr{C}^{1}_{\pm}$) with equilibria lead to
either a heteroclinic cycle $\Lambda$ or four different homoclinic cycles
($\Lambda_{\pm}$ and $\Gamma_{\pm}$). Each pair of limit cycles
$(\mathscr{C}^{2}_{\pm},\mathscr{C}^{1}_{\pm})$ may respectively merge
together and disappear. This is a saddle-node bifurcation of limit cycles.
Different combinations of these give rise to a rich list of bifurcation
scenarios. Finite determinacy of each of these bifurcations has been
thoroughly investigated. Subcritical and supercritical types of bifurcations
can be switched using small changes into controller coefficients. This greatly
influences their stabilization potential in applications. Our symbolic
estimation of critical controller sets provide a computationally feasible
approach for bifurcation control of such systems. We derive novel estimates
for the heteroclinic, homoclinic and limit cycles to facilitate the amplitude
size management and frequency control of the nearby oscillating dynamics. To
illustrate our approach, we consider Chua system with a quadratic state-
feedback controller. Our approach provides estimated controller sets in terms
of the original controller coefficients and constants of the controlled Chua
system. Controlled Chua system experiences a pitchfork bifurcation, three Hopf
bifurcations and two homoclinic bifurcations. We show that there exist two
different regions of controller coefficient choices for feedback
regularization and two nearby regions for supercritical Hopf stabilization
approach.

Keywords: Singular control; Critical controller sets; Uncontrollable
linearization; Subcritical and supercritical switching.

2010 Mathematics Subject Classification: Primary: 34H20, 34K18, 34C20;
Secondary: 58E25.

##  1 Introduction

Differential systems with symmetry (equivariant systems) frequently occur in
many real life and engineering problems while qualitative changes are the
intrinsic elements of their evolutions. A system is called singular when it
experiences a qualitative change and each qualitative change is called a
bifurcation. Hence, bifurcation control of equivariant singular systems is a
natural contribution for the management of their qualitative evolutions. Due
to the singularity around any qualitative change, an uncontrolled system may
potentially experience varieties of desired and undesired dynamics. We
distinguish different bifurcation scenarios for potential controlled dynamics,
where they can be realised and/or switched as desired through a state-feedback
singular bifurcation controller with small controller coefficients. These
bifurcations can be quantitatively controlled, prevented, delayed or
accelerated. Hence, the controlled system can experience any desired dynamics
chosen from a rich list of bifurcation scenarios through tuning small
controller coefficients. We refer to these by bifurcation control problem.
Singularity lays an asset and important potential for engineering applications
with high manoeuvring capability. Manoeuvrability here implies frequent
quantitative and qualitative dynamics changes with minimal controller costs.
Our proposed approach makes a full use of the internal singular dynamics of
the uncontrolled system to enforce the desired dynamics. This is an
alternative to many existing techniques in nonlinear control theory such as
back-stepping method, input-state feedback linearization, Lyapunov functions,
etc; e.g., see [39]. Most of these techniques are oblivious of the
uncontrolled singular dynamics: the designed controllers (fully or partially)
eliminate the internal (uncontrolled) dynamics and replace it with an already-
known desired and non-singular dynamics. Thus, they mainly fail to exploit the
benefits of singularities; this includes the manoeuvring capabilities. The
main obstacle originates from the underlying complexity (highly rich dynamics)
of singular systems. Bifurcation control stands to efficiently make use of the
intrinsic singular dynamics of the uncontrolled system. This justifies to call
our proposed controller approach as singular control. Therefore, bifurcation
control leads to an effective approach with low-cost controllers and high
manoeuvrability.

Three main claimed contributions in this paper are as follows: (1) Complete
symmetry-breaking classification for the highly rich bifurcation scenarios
associated with $\mathbb{Z}_{2}$-equivariant Bogdanov-Takens singularity. (2)
Novel symbolic estimates of critical controller sets, where they are
sufficiently accurate for many of their potential applications. (3) An
introduction of a practically feasible approach for singular control of
linearly uncontrollable systems with arbitrary state dimension and two zero
eigenvalues for its (non-hyperbolic) linearization.

There are a rich list of bifurcation scenarios where they can all be realised
through our proposed approach. These are only useful when the controller
design is adaptable based on the physics of the problem. This is, of course,
one of our main claimed contributions. Furthermore, bifurcations may switch
their subcritical type with supercritical types (or vice versa) when small
changes are applied to the controller coefficients; see Theorem 3.7 and Remark
4.3. These signify the importance for the study of symmetry breaking
bifurcations due to their influence for the stabilizing approach in
applications. Small modeling imperfections for singular systems lead to
bifurcations and thus, they can be a dominant factor for determining the
dynamics. For $\mathbb{Z}_{2}$-equivariant systems, bifurcations include the
loss of symmetry; this is technically called a symmetry breaking bifurcation.
Hence, equivariant bifurcation control is not sufficient for systems whose
noises and imperfections have the potentials for symmetry breaking. Thus,
symmetry breaking bifurcation analysis and control for singular systems is
necessary in these cases. Uncontrolled smooth differential systems whose
linearization at a non-hyperbolic equilibrium has a pair of zero-eigenvalues
(none semi-simple and $\mathbb{Z}_{2}$-equivariant mode cases) can be reduced
to

$\displaystyle\frac{dx}{dt}=f(x,y),\;\frac{dy}{dt}=a_{0}x+g(x,y),\;f(-x,-y)=-f(x,y),\;g(-x,-y)=-g(x,y),$
(1.1) $\displaystyle\hbox{ for all }x,y\in\mathbb{R},a_{0}\neq 0,\hbox{ and
}f(0,0)=g(0,0)=0.$

Functions $f$ and $g$ are assumed to be $C^{k}$-smooth for $k\geq 5$. The
state space of the original system of such types can be any arbitrarily large
natural number and it can yet be reduced into the differential system (1.1).
This is feasible through Jordan canonical transformation and a center manifold
reduction; e.g., see Section 5 where this has been implemented to Chua system
with a three dimensional state space. Hence, the bifurcation control problem
for such $\mathbb{Z}_{2}$-equivariant systems (using center manifold
reductions, also see[27]) through polynomial controllers $(u_{1},u_{2})$ can
be reduced to

$\displaystyle\dot{x}=\frac{dx}{dt}=f(x,y)+u_{1}(x,y,\nu),\;\dot{y}=\frac{dy}{dt}=a_{0}x+g(x,y)+u_{2}(x,y,\nu),\;\,u_{1}(x,y,\mathbf{0})=u_{2}(x,y,\mathbf{0})=0,$
(1.2)

for $\nu=(\nu_{1},\ldots,\nu_{m})\in\mathbb{R}^{m}.$ We call $\nu_{j}$ (for
$j\leq m$) by controller coefficients. We further assume that

$\displaystyle\left({\frac{\partial^{3}f}{\partial
x^{3}}(0,0)}\right)\left(\frac{1}{4}\frac{\partial^{3}f}{\partial x\partial
y^{2}}(0,0)+3\frac{\partial^{3}g}{\partial y^{3}}(0,0)\right)\neq 0.$

Then, there are always locally invertible polynomial changes of state
variables to transform system (1.2) into (see Remark 2.1)

$\displaystyle\dot{x}=\mu_{0}+\mu_{1}y+\mu_{2}x+a_{1}y^{3}+b_{0}xy^{2}+\sum_{l=1}^{\lfloor\frac{N-1}{4}\rfloor}(b_{l}+\mu_{2l+2})xy^{4l}+\sum_{l=0}^{\lfloor\frac{N-2}{4}\rfloor}\mu_{2l+3}xy^{4l+1},$
$\displaystyle\dot{y}=-x+\mu_{2}y+b_{0}y^{3}+\sum_{l=1}^{\lfloor\frac{N-1}{4}\rfloor}(b_{l}+\mu_{2l+2})y^{4l+1}+\sum_{l=0}^{\lfloor\frac{N-2}{4}\rfloor}\mu_{2l+3}y^{4l+2},$
(1.3)

modulo higher degree terms than $N$ where each $\mu_{i}$ is a polynomial in
terms of controller coefficients $\nu,$
$a_{1}:=\frac{1}{3!}\frac{\partial^{3}f}{\partial x^{3}}(0,0),$
$b_{0}:=\frac{1}{8}\frac{\partial^{3}f}{\partial x\partial
y^{2}}(0,0)+\frac{1}{2}\frac{\partial^{3}g}{\partial y^{3}}(0,0)$ and
$b_{i-4}$ (when $i>4$); also see [18, Section 5.2]. When
$(\mu_{0},\mu_{1},\mu_{2},\mu_{3})$ is invertible, $m=4$ and $\mu_{i}$-s for
$i=0,1,2,3$ can play the same role as controller coefficients $\nu$. Two
parameters are sufficient to unfold the system for symmetry-preserving
bifurcation control of (1.3). However, all four parameters are needed for
fully unfolding the system to include both symmetry-preserving and symmetry-
breaking bifurcations. In other words, system (1.3) is a codimension-four
universal unfolding system. This means that the system has enough controller
coefficients in the right places to disclose and enforce all potential local
bifurcation scenarios from the differential system (1.2). Then, controller
coefficients dominate possible modeling imperfections and small noises. The
origin is always an equilibrium as a primary equilibrium when $\mu_{0}=0$. The
primary equilibrium is deviated from the origin when $\mu_{0}$ ($|\mu_{0}|\ll
1$) is slightly varied. Two new equilibria bifurcate from the origin and thus,
they are called by secondary equilibria $E_{\pm}.$ Limit cycles
$\mathscr{C}^{1}_{\pm}$ and $\mathscr{C}^{2}_{\pm}$ are bifurcated through
Hopf and Bautin bifurcations from the secondary equilibria and are called as
tertiary limit cycles. Hence, homoclinic and heteroclinic cycles are
quaternary cycles and their bifurcations are called quaternary bifurcations. A
homoclinic cycle is a trajectory that joins an equilibrium to itself as the
red cycle in Figure 3(c). A heteroclinic cycle here consists of two
trajectories where they connect two different equilibria; see the red cycle in
Figure 3(d). Critical controller sets in terms of controller coefficients
distinguish these bifurcations and determine where they precisely occur. Small
size neighborhood validity of the secondary, tertiary, and quaternary
bifurcations reduces potential applications of the theory in practical life
problems. So, it is important to derive and present high order symbolic
estimates for critical controller sets.

Bifurcation theory has an extensive literature of more than a century; e.g.,
see [23, 7, 32, 10]. However, their application in control engineering has had
a slow progress due to its technical challenges. Most theoretical results
address possible bifurcation scenarios. However, they fail to precisely locate
the desired dynamics in terms of the original parameters and constants of
practical life problems. Bifurcation control has recently attracted some
researchers and includes important contributions; see [20, 21, 17, 42, 8, 9,
30, 31, 26, 25, 18]. Kang [30] considered control systems with a single input
and characterized generic normal form systems by their quadratic invariants
and the equilibrium sets. Kang et al [31] considered a singular system with
uncontrollable linearization that includes a single uncontrollable mode (a
zero eigenvalue). They showed that there are always nearby bifurcated
equilibria around the origin, where they are controllable. This is a
significant contribution as many of approaches in nonlinear control cannot be
applied to systems with uncontrollable linearizations. Our introduced approach
in this paper can be considered as a generalization of Kang et al [31] to
include uncontrollable modes with two zero eigenvalues. Wu and Yu [42]
proposed a method to delay static and dynamic bifurcations. Hamzi et al [26]
considered a family of singular control systems with two purely imaginary
uncontrollable modes. They presented state-feedback controller for their
stabilization and the quadratic invariants characterizing the generic Hopf
bifurcation control. Hamzi [25] considered the singular control families with
double zero uncontrollable modes. He obtained the quadratic invariants of the
family. The quadratic invariants are used for synthesizing a quadratic
stabilizing controller and for characterizing the generic normal forms. Our
results generalizes his results for a degenerate normal form family and
symmetry-breaking cases, where we addresses full singular bifurcation control
problem. We studied bifurcation control for families of Bogdanov-Takens
singular systems including a $\mathbb{Z}_{2}$-equivariant family in [18]. In
this paper we skip the existing dynamics that are implied from our equivariant
dynamical analysis in [18]; see Remark 3.12. Further, we have investigated the
bifurcation control for a singular family on a three dimensional central
manifold with two imaginary uncontrollable modes in [17]. Gazor and Shoghi
[20, 19] considered applications of bifurcation control for sound intensities
in music; also see [21, 23, 7, 32, 6, 38]. An efficient nonlinear time
transformation method has been recently developed in [35] and applied to
obtained highly accurate estimates for global bifurcations of homoclinic and
heteroclinic varieties of codimension two singularities. We employ a novel
generalisation of their approach to deal with our codimension four controlled
system; also see [36, 1, 2, 3, 38, 6] for closely related results and
techniques.

This draft is organized as follows. Jet sufficiency of equilibrium
bifurcations are discussed in Section 2. Finite determinancy In section 3, we
study local symmetry breaking bifurcations for controlled system (1.3). We
derive and present a rich list of controller manifolds in terms of controller
coefficients in this section. These give rise to an effective tool not only to
cause or prevent bifurcations but also to satisfy control objectives such as
feedback regularization and stabilization techniques via supercritical Hopf
and homoclinic/heteroclinic bifurcations. Further, we derive leading estimates
for homoclinic and heteroclinic cycles as well as the amplitudes and angular
frequencies of bifurcated limit cycles in terms of controller coefficients.
These provide efficient criteria for the amplitude size control and frequency
management of the nearby oscillating dynamics. Bifurcation control of
$\mathbb{Z}_{2}$-equivariant systems is considered in section 4 using a single
input quadratic controller. We show how a system with an uncontrollable
linearization can be treated through our proposed symmetry-breaking
bifurcation control. We show that the controlled system admits two saddle node
controller manifolds, two Hopf controller sets (one is supercritical and the
other is subcritical) and two homoclinic controller sets for linearly
uncontrollable case; see Theorem 4.2 and its proof. Section 5 is dedicated to
illustrate our results on Chua differential system. This system in the
vicinity of its Bogdanov-Takens singularity may undergo a pitchfork
singularity, three Hopf bifurcations and three homoclinic bifurcations. They
can be all realised through an input state-feedback controller design. We
explore feedback regularization of the origin and state feedback stabilization
via supercritical Hopf bifurcations. We prove that there exist two regions of
controller coefficient choices for feedback regularization while feedback
stabilization approach admits controller coefficient choices from other two
regions. Freedoms of choices for controller coefficients within these regions
facilitate the amplitude size and frequency managements of the oscillating
dynamics. We show how a very small single-input controller readily enforces
all these control objective; e.g., see figures 9(b), 9(c) and 9(d). Finally,
conclusions are drawn in section LABEL:secConclusion.

##  2 Finite determinacy for bifurcations of equilibria

The infinite Taylor expansion of normal forms is an obstacle both in the
theoretical analysis in bifurcation theory and in the practical computations
using computer algebra systems; e.g., see [17, 11, 12, 10]. One usually
truncates the infinite normal forms up to certain degree $k$; this is called a
$k$-jet of the differential system. In other words, one ignores the higher
order terms in normal form expansion. Higher order terms are never derived due
to the complexity of formulas and impractical computations using computers.
Hence, the qualitative equivalence of the truncated normal forms and the
original normal forms is an important question and needs to be thoroughly
investigated. The bifurcation analysis of the truncated normal forms may be
inaccurate, misleading and/or essentially wrong when the question of jet
sufficiency is not investigated.

Consider an equivalence relation and recall that a property is defined as a
qualitative property when it is invariant under the equivalence relation. When
a truncated normal form reflects the qualitative dynamics of the original
system, we refer to the system as a finitely determined system. The jet
sufficiency refers to the degree upon which the truncated system is sufficient
to fully represent the qualitative dynamics of the original system. Different
equivalence relations are necessary depending on the intended properties in
our analysis. We first consider contact equivalence relation and appeal to
singularity theory. Contact equivalence is the finest equivalence relation for
equilibrium bifurcations of a given vector field. The results from singularity
theory are compatible with normal forms of various types, i.e., normal forms,
orbital normal forms, parametric normal forms, and also concepts such as
universal asymptotic unfolding, etc; see [17]. We follow [24, 13] and consider
one of the parameters as a distinguished parameter and denote it by $\lambda$.
Two mappings $f(x,\lambda)$ and $g(x,\lambda)$ are contact equivalent if there
exists a local diffeomorphism germ $X(x,\lambda)$ with $X(0,0)=0,$ locally
invertible map $\Lambda(\lambda),$ $\Lambda(0)=0,$ and a locally nonsingular
$n\times n$ matrix $S(x,\lambda)$ such that
$f(x,\lambda)=S(x,\lambda)g(X(x,\lambda),\Lambda(\lambda))$; see [24, page
166] and [13].

######  Remark 2.1.

Any $\mathbb{Z}_{2}$-equivariant differential system (1.1) can be transformed
into a $\mathbb{Z}_{2}$-equivariant differential system using permissible
$\mathbb{Z}_{2}$-invariant polynomial transformations. Then, the remaining
terms in the normal form system will be the $\mathbb{Z}_{2}$-equivariant terms
from those remaining in the normal form system in [18, Thoerem 2.9]. As
similar to [18, Theorem 3.3], any multiple parametric perturbation (1.2)
(including $\mathbb{Z}_{2}$-breaking perturbation terms) of the
$\mathbb{Z}_{2}$-equivariant differential system (1.1) can be transformed into
(1.3).

Since we are dealing with local bifurcations of vector fields, we define germs
of vector fields at the origin. Two vector fields are defined as germ
equivalent when there is a neighborhood where they are equal on that. Then,
each equivalent class of vector fields from germ equivalent relation is called
a germ vector field. The steady-state bifurcation problem associated with the
normal form system of the generalized cusp case of Bogdanov-takens singularity
is given by $G(x,y,\lambda):=(G_{1},G_{2})=(0,0)$ where
$G_{1}(0,0,0)=G_{2}(0,0,0)=0$,

$\displaystyle(G_{1},G_{2}):=\left(a_{1}y^{3}+b_{0}xy^{2}+\lambda+\sum_{l=1}^{\lfloor\frac{N-1}{4}\rfloor}b_{l}xy^{4l},a_{0}x+b_{0}y^{3}+\sum_{l=1}^{\lfloor\frac{N-1}{4}\rfloor}b_{l}y^{4l+1}\right)+h.o.t.,$
(2.1)

$\lambda:=\mu_{0},$ $a_{0}a_{1}b_{0}\neq 0$; see [18] for more details.

######  Theorem 2.2.

The germ $G$ is contact equivalent with $G+p$ for all
$p\in\overrightarrow{\mathcal{M}}^{4}$.

###### Proof.

Normal form results and the approach in singularity theory are compatible. The
argument relies on the fact that smooth changes of coordinates, time rescaling
and reparametrization all are compatible with contact equivalence relation and
transform a germ to its contact equivalent germ. Furthermore, formal normal
forms can be extended into smooth cases using Borel lemma modulo flat parts.
We follow [24, Definition 7.1, Proposition 1.4, Theorem 7.2 and Theorem 7.4]
and instead prove that
$\overrightarrow{\mathcal{M}}^{4}\subseteq\mathcal{K}(G),$ where
$\overrightarrow{\mathcal{M}}$ is the generated module

$\overrightarrow{\mathcal{M}}:=\bigg{<}{x\choose 0},{0\choose x},{y\choose
0},{0\choose y},{\lambda\choose 0},{0\choose\lambda}\bigg{>}$ (2.2)

over $\mathscr{E}_{x,y,\lambda}.$ Thus, $\overrightarrow{\mathcal{M}}^{4}$ is
the space of all smooth vector field germs whose Taylor expansions do not have
monomial vector field terms of degree less than 4. Here,
$\mathscr{E}_{x,y,\lambda}$ is the local ring of all smooth germs in
$(x,y,\lambda)$-variables. Define its unique maximal ideal by
$\mathcal{M}:=<x,y,\lambda>$. Note that the flat vector fields live in
$\overrightarrow{\mathcal{M}}^{3}$. Define an
$\mathscr{E}_{x,y,\lambda}$-module $\mathcal{K}(G)$ that is generated by

$\mathcal{M}^{2}{G_{1x}\choose G_{2x}},\mathcal{M}^{2}{G_{1\rho}\choose
G_{2\rho}},\mathcal{M}{G_{1}\choose 0},\mathcal{M}{G_{2}\choose
0},\mathcal{M}{0\choose G_{1}},\mathcal{M}{0\choose G_{2}}.$ (2.3)

When ${\overrightarrow{\mathcal{M}}}^{k+1}\subseteq\mathcal{K}(f)$, by [24,
Theorem 7.2], $G$ is $k$-sufficient with respect to contact equivalence
relation. Now we choose $a_{1}:=1$ to simplify the formulas. Then, define

$J:=\bigg{<}{0\choose
xy}\bigg{>}+{\mathcal{\overrightarrow{\mathcal{M}}}}^{4}.$

We first recall Nakayama Lemma. For any $\mathscr{E}_{x,y,\lambda}$-modules
$J$ and $\mathcal{K},$ the Nakayama Lemma implies that $J\subseteq\mathcal{K}$
if and only if $J\subseteq\mathcal{K}+\mathcal{M}J.$ Thereby, we denote
$\simeq$ for equalities modulo terms in $\mathcal{M}J$:

$\lambda^{i}x^{j}y^{k}{G_{1}\choose 0}\simeq{\lambda^{i+1}x^{j}y^{k}\choose
0},\hskip 14.22636pt\lambda^{i}x^{j}y^{k}{0\choose
G_{1}}\simeq{0\choose\lambda^{i+1}x^{j}y^{k}},\\\ $

where $i+j+k=3,$ $i,j,k\geq 0.$ These conclude membership of
${\lambda^{4}\choose 0},$ ${\lambda x^{3}\choose 0},$ ${\lambda x^{2}y\choose
0},$ ${\lambda xy^{2}\choose 0},$ ${\lambda y^{3}\choose 0},$
${0\choose\lambda^{4}},$ ${0\choose\lambda x^{3}},$ ${0\choose\lambda
x^{2}y},$ ${0\choose\lambda xy^{2}},$ and ${0\choose\lambda
y^{3}}\in\mathcal{K}_{s}+\mathcal{M}J.$ Next, we consider

$\displaystyle x^{3}{G_{2}\choose 0}\simeq{-x^{4}\choose 0},\quad
y^{3}{G_{2}\choose 0}\simeq{-xy^{3}\choose 0},\quad x^{3}{0\choose
G_{2}}\simeq{0\choose-x^{4}},\quad y^{3}{0\choose G_{2}}\simeq{0\choose-
xy^{3}}.$

Similarly, we can show that ${x^{i+1}y^{j}\choose 0},{0\choose
x^{i+1}y^{j}}\in\mathcal{K}_{s}+\mathcal{M}J$ for $i+j=3$ and $i,j\geq 0$. In
the one hand, we have

$x^{i}y^{j}{G_{2}\choose 0}\simeq{x^{i+1}y^{j}\choose 0},\hskip
14.22636ptx^{i}y^{j}{0\choose G_{2}}\simeq{0\choose x^{i+1}y^{j}}$

Since ${xy^{3}\choose 0}\in\mathcal{K}+\mathcal{M}J$ and

$xy{G_{1x}\choose G_{2x}}\simeq{b_{0}xy^{3}\choose-xy},\quad{0\choose
xy}\in\mathcal{K}+\mathcal{M}J.$

On the other hand,

$y{0\choose G_{2}}={0\choose-xy+b_{0}y^{4}}\quad\hbox{ implies that
}\quad{0\choose y^{4}}\in\mathcal{K}+\mathcal{M}J.$

Finally, $y^{2}{G_{1y}\choose G_{2y}}\simeq{y^{4}\choose 0}$ completes the
proof. ∎

##  3 Symmetry breaking and critical controller sets

Bifurcation control is facilitated by introducing critical controller sets or
controller manifolds. Critical controller sets are typically codimension-one
bifurcation manifolds within the controller coefficient space. This is a
necessary condition of critical controller manifolds to provide a partition
for the controller coefficient space into a finite number of connected
regions. When controller coefficients from critical controller sets are
subjected to small perturbations, the qualitative dynamics of the controlled
system changes. The neighbourhood validity of controller sets is greatly
influenced by the relative geometry of these manifolds in four-dimension. For
example, a limit cycle $\mathscr{C}_{0}$ may collide with equilibrium $E_{+}$
and disappear at a homoclinic controller set $T_{HmC+}.$ It can alternatively
collide with another equilibrium $E_{-}$ and disappear at different homoclinic
controller set $T_{HmC-}$. Therefore, controller manifold $T_{HmC+}$ is no
longer valid if $\mathscr{C}_{0}$ has already been disappeared through
$T_{HmC-},$ or vice versa. We provide local criteria for our derived critical
controller manifolds. These criteria are helpful and necessary for both
deriving critical controller sets and the distinction of the neighbourhood
validity within the controller coefficient space. Further, we apply
alternative normalized systems and truncation degrees in our formulation. We
derive the bifurcation controller sets in terms of symbolic constants and
unknown controller coefficients. This is important for many control
engineering applications.

Equation (1.3) is $\mathbb{Z}_{2}$-equivariant for $\mu_{0}=\mu_{3}=0$; see
[18] for $\mathbb{Z}_{2}$-equivariant bifurcation control. The symmetry-
breaking occurs when either of $\mu_{3}=0$ and $\mu_{0}=0$ or both fail. Due
to the geometric complexity of bifurcation controller sets in four dimensional
space and errors of estimates, bifurcation controller sets are generally valid
within certain neighborhoods. We provide these sets under two categories:
symmetry breaking analysis using $\mu_{3}$ (while $|\mu_{0}|\ll 1$) and
symmetry-breaking analysis through $\mu_{0}$ and $\mu_{3}$ for
$\mu_{0}\mu_{3}\neq 0$. Here, we use the notation of the little  $\mathscr{O}$
for parameters, where they are generally polynomial functions of the original
control parameters.

######  Proposition 3.1.

For
$|\mu_{0}|={\scalebox{0.7}{$\mathscr{O}$}}(||(\mu_{1},\mu_{2},\mu_{3})||^{4}),$
estimated primary pitchfork and Hopf bifurcation controller sets are

$T_{P}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{1}+{\mu_{2}}^{2}=0\right\\}\;\hbox{
and
}\;T_{H}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{2}=0,\mu_{1}>0\right\\}.$
(3.1)

  * •

[18, Theorem 5.1]: We have a super-critical Hopf bifurcation when $b_{0}<0.$
Bifurcated limit cycle $\mathscr{C}_{0}$ appears when $\mu_{2}>0$ and is
asymptotically stable. For $b_{0}>0,$ the system undergoes a sub-critical Hopf
bifurcation and the limit cycle $\mathscr{C}_{0}$ exists when $\mu_{2}<0$ and
is unstable. There is a local bifurcation of secondary equilibria $E_{\pm}$
from the origin (or an equilibrium slightly deviated from the origin) in
nearby of $T_{p}.$

  * •

Estimated radius and angular frequency for the bifurcated limit cycles
$\mathscr{C}_{0}$ are given by $\sqrt{-\frac{2\mu_{2}}{b_{0}}}$ and
$\sqrt{-\mu_{1}}.$ These are useful for magnitude and frequency management of
the oscillating dynamics. There are a bifurcation controller set $\mathscr{B}$
and an estimated hysteresis controller set $\mathscr{H}$ given by

$\displaystyle\mathscr{B}:=\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{0}=0\\}\hbox{
and
}\mathscr{H}:=\big{\\{}(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{0}=\frac{8{\mu_{3}}^{3}{(-\mu_{1})}^{\frac{3}{2}}}{27(a_{1}+2b_{0}\sqrt{-\mu_{1}})^{2}}\big{\\}}.$
(3.2)

Critical controller sets $\mathscr{B}$ and $\mathscr{H}$ contribute into the
symmetry-breaking and bifurcation diagram classification of the pitchfork
singularity for controlled system (1.3); see Figure 1.

  * •

A symbolic approximation for the secondary equilibria of system (1.3) follows

$\displaystyle
E_{\pm}:\quad(x_{\pm},y_{\pm}):=\left(\mu_{2}y_{\pm}+\mu_{3}{y_{\pm}}^{2}+b_{0}{y_{\pm}}^{3},\dfrac{-\mu_{2}\mu_{3}\pm\sqrt{-a_{1}\mu_{1}-a_{1}{\mu_{2}}^{2}-2b_{0}\mu_{1}\mu_{2}-2b_{0}{\mu_{2}}^{3}-\mu_{1}{\mu_{3}}^{2}}}{a_{1}+2b_{0}\mu_{2}+{\mu_{3}}^{2}}\right)\\!.$
(3.3)

###### Proof.

The eigenvalues of the linearised system at the origin are
$\mu_{2}\pm\sqrt{-\mu_{1}},$ where we treat $\mu_{0}$ as a small perturbation.
Hence, equation (3.1) is a Hopf bifurcation manifold and $\sqrt{-\mu_{1}}$
stands for the leading term in the angular velocity of the oscillating
dynamics. Consider
$|\mu_{2}|,|\mu_{3}|={\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|).$ Note that
parameters $\mu_{2}$ and Then, a normalized amplitude equation in polar
coordinates $(\rho,\theta)$ is
$\dot{\rho}=\mu_{2}\rho+\frac{b_{0}}{2}\rho^{3}+{\scalebox{0.7}{$\mathscr{O}$}}(\rho^{5},|\mu_{1}|^{2}).$
An estimated radius for the bifurcated limit cycle from this system is
$\sqrt{-\frac{2\mu_{2}}{b_{0}}}$; see proof of [18, Theorem 5.1]. Thus, the
radius of the bifurcated limit cycle grows when $\frac{\mu_{2}}{b_{0}}$
decreases. Further, we have a single zero singularity when $\lambda_{1}=0$ for
$\lambda_{0}:={\mu_{2}}-\sqrt{-\mu_{1}}$; i.e., $\mu_{1}\leq 0.$ Then, a
truncated and re-scaled differential equation on the center manifold (also see
[27]) is given by

$\dot{x}=\mu_{0}+\lambda_{0}\sqrt{-\mu_{1}}x+\frac{\sqrt{-\mu_{1}}\mu_{3}}{2}x^{2}+\left(\frac{b_{0}{\sqrt{-\mu_{1}}}}{4}+\frac{a_{1}}{8}\right)x^{3}+{\scalebox{0.7}{$\mathscr{O}$}}(||(\mu,x)||^{4}).$
(3.4)

Now we appeal to singularity theory developed in [24]. By [24, Proposition
4.4], this equation is a universal unfolding for the pitchfork singularity.
Critical set $\mathscr{B}$ and hysteresis $\mathscr{H}$ follow [24, Pages
140]. Treat $\lambda_{0}$ as the distinguished bifurcation parameter and
$G(x,\lambda_{0},\mu_{0},\mu_{1},\mu_{3}):=\mu_{0}+\lambda_{0}\sqrt{-\mu_{1}}x+\frac{\sqrt{-\mu_{1}}\mu_{3}}{2}x^{2}+\left(\frac{b_{0}{\sqrt{-\mu_{1}}}}{4}+\frac{a_{1}}{8}\right)x^{3}$.
Hence, the bifurcation set $\mathscr{B}$ is obtained from $G=\frac{\partial
G}{\partial x}=\frac{\partial G}{\partial\lambda_{0}}=0$ while the hysteresis
controller set $\mathscr{H}$ is given by $G=\frac{\partial G}{\partial
x}=\frac{\partial^{2}G}{{\partial x}^{2}}=0.$ Here,

$\frac{\partial G}{\partial
x}=\lambda_{0}\sqrt{-\mu_{1}}+\mu_{3}\sqrt{-\mu_{1}}x+3\left(\frac{b_{0}}{4}\sqrt{-\mu_{1}}+\frac{a_{1}}{8}\right)x^{2},$

$\frac{\partial G}{\partial\lambda_{0}}=x\sqrt{-\mu_{1}},$ and
$\frac{\partial^{2}G}{{\partial
x}^{2}}=\mu_{3}\sqrt{-\mu_{1}}+3(\frac{b_{0}}{2}\sqrt{-\mu_{1}}+\frac{a_{1}}{4})x.$
Omitting $x$ and $\lambda_{0}$ from these equations gives rise to the
governing equations for $\mathscr{B}$ and $\mathscr{H}$. To estimate the
secondary equilibria $E_{\pm}$, we derive $x$ from the second steady-state
equation corresponding with (1.3). Then, we substitute $x$ into the first
steady-state equation to obtain
$(2b_{0}\mu_{2}+{\mu_{3}}^{2}+a_{1})y^{3}+2\mu_{2}\mu_{3}y^{2}+(\mu_{1}+{\mu_{2}}^{2})y=0$
modulo $\mu_{0}$ and terms of degree four and higher in $y$. Roots of this
cubic polynomial are $y=0$ and $y_{\pm}$ in (3.3). Hence, the second steady-
state equation for (1.3) concludes the formula (3.3). ∎

######  Theorem 3.2 (Saddle-node controller sets).

There are two saddle-node bifurcations at critical controller sets estimated
by $T^{\pm}_{SN}=\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\xi_{\pm}=0\\}$ where

$\displaystyle\xi_{\pm}:=\mu_{0}-\frac{\mp
2\left(6b_{0}\mu_{1}\mu_{2}+3a_{1}\mu_{1}+3{\mu_{3}}^{2}\mu_{1}+3a_{1}{\mu_{2}}^{2}-{\mu_{2}}^{2}{\mu_{3}}^{2}+6b_{0}{\mu_{2}}^{3}\right)\sqrt{{\mu_{2}}^{2}{\mu_{3}}^{2}-6b_{0}{\mu_{2}}^{3}-3{\mu_{3}}^{2}\mu_{1}-3a_{1}{\mu_{2}}^{2}-6b_{0}\mu_{1}\mu_{2}-3a_{1}\mu_{1}}}{27\left(a_{1}+{\mu_{3}}^{2}+2b_{0}\mu_{2}\right)^{2}}$
$\displaystyle+\frac{2\left({\mu_{2}}^{3}{\mu_{3}}^{3}+18b_{0}{\mu_{2}}^{4}\mu_{3}+9\mu_{1}\mu_{2}{\mu_{3}}^{3}+18b_{0}\mu_{1}{\mu_{2}}^{2}\mu_{3}+9a_{1}\mu_{1}\mu_{2}\mu_{3}+9a_{1}{\mu_{2}}^{3}\mu_{3}\right)}{27\left(a_{1}+{\mu_{3}}^{2}+2b_{0}\mu_{2}\right)^{2}}.$
(3.5)

###### Proof.

The characteristic polynomial coefficients of $\lambda^{2}+d_{1}\lambda+d_{2}$
for Jacobian matrix of (1.3) follows
$d_{1}\\!:=\\!4b_{0}y^{2}+3\mu_{3}y+2\mu_{2},$ and

$\displaystyle
d_{2}\\!:=\\!5{b_{0}}^{2}y^{4}+8b_{0}\mu_{3}{y}^{3}+3\left({\mu_{3}}^{2}+2b_{0}\mu_{2}+a_{1}\right)y^{2}+4\mu_{2}\mu_{3}y+{\mu_{2}}^{2}+\mu_{1}.\quad$
(3.6)

The scalars $d_{1}$ and $d_{2}$ are arrays of the first column of Routh table
(e.g., see [41]). We claim that there are two local equilibria, where they
undergo saddle-node bifurcations. Let $(x,y)=(x_{0},{y_{0}})$ be one of these
two. Then, $(x_{0},{y_{0}})$ must satisfy $d_{2}(y_{0})=0$ and the steady-
state equations of (1.3). These provide a precise implicit formulation for the
saddle-node singularity. Shift of coordinates $Y=y-{y_{0}}$ and $X=x-x_{0}$
give rise to

$\displaystyle\dot{X}\\!=\\!\left(\mu_{3}{{y_{0}}}+b_{0}{{y_{0}}}^{2}+\mu_{2}\right)\\!X\\!+(b_{0}{{y_{0}}}\mu_{2}+3a_{1}{{y_{0}}}+b_{0}\mu_{3}{{y_{0}}}^{2}+{{y_{0}}}^{3}{b_{0}}^{2}+b_{0}X){Y}^{2}+(\mu_{3}+2b_{0}{{y_{0}}})XY$
$\displaystyle+(\mu_{1}+\mu_{3}{{y_{0}}}\mu_{2}+2b_{0}{{y_{0}}}^{2}\mu_{2}+3a_{1}{{y_{0}}}^{2}+3\mu_{3}{{y_{0}}}^{3}b_{0}+2{{y_{0}}}^{4}{b_{0}}^{2}+{\mu_{3}}^{2}{{y_{0}}}^{2})Y+a_{1}{Y}^{3}+\xi_{\pm},$
$\displaystyle\dot{Y}=(\mu_{2}+2\mu_{3}{{y_{0}}}+3b_{0}{{y_{0}}}^{2})Y-X+(\mu_{3}+3b_{0}{{y_{0}}})Y^{2}+b_{0}Y^{3}.$
(3.7)

The eigenvalues $\lambda_{\pm}$ of Jacobian are then given by

$\displaystyle\mu_{2}+\frac{3}{2}\mu_{3}{y_{0}}+2b_{0}{{y_{0}}}^{2}\pm\frac{\mathbf{i}\sqrt{4{y_{0}}\mu_{2}\mu_{3}+8b_{0}{{y_{0}}}^{2}\mu_{2}+3{\mu_{3}}^{2}{{y_{0}}}^{2}+8{{y_{0}}}^{3}b_{0}\mu_{3}+4{{y_{0}}}^{4}{b_{0}}^{2}+12a_{1}{{y_{0}}}^{2}+4\mu_{1}}}{2}.$

When
${\emph{sign}}(\mu_{2}+\frac{3}{2}\mu_{3}{y_{0}}+2b_{0}{{y_{0}}}^{2}\pm\frac{1}{2})=\pm
1,$ there is a polynomial
$h_{\pm}(z,\xi_{\pm}):=\gamma_{0}\xi_{\pm}+\gamma_{1}z^{2}+\gamma_{2}\xi_{\pm}^{2}+\gamma_{3}z\xi_{\pm},$
$z\in\\{X,Y\\},$ for a quadratic approximation of the center manifold. We here
assume that $|\xi_{\pm}|\ll|y_{0}|.$ We apply the center manifold reduction
procedure to obtain $\gamma_{0}=-\frac{1}{4b_{0}{{y_{0}}}^{2}},$
$\gamma_{1}=\frac{2{b_{0}}^{2}{{y_{0}}}^{2}-3a_{1}}{64{b_{0}}^{3}{y_{0}}^{5}},$
$\gamma_{2}=0,$ and
$\gamma_{3}=\frac{4{b_{0}}^{2}{{y_{0}}}^{2}-3a_{1}}{64{b_{0}}^{4}{y_{0}}^{7}}.$
By a time-rescaling, the governing differential equation on the center
manifold follows

$\displaystyle\dot{Y}=\big{(}\frac{5}{8}{y_{0}}^{2}+\frac{3a_{1}}{16{b_{0}}^{2}}\big{)}Y^{2}+\xi_{\pm}\big{(}\frac{3}{16b_{0}}+\frac{3a_{1}}{32{b_{0}}^{3}{{y_{0}}}^{2}}\big{)}Y+\xi_{\pm}\big{(}{y_{0}}^{3}+\frac{3a_{1}{\xi_{\pm}}}{256{b_{0}}^{4}{y_{0}}^{4}}+\frac{\xi_{\pm}}{128{b_{0}}^{2}{y_{0}}^{2}}\big{)}.$
(3.8)

The discriminant is given by
$\frac{\xi_{\pm}\left(\xi_{\pm}-48a_{1}{y_{0}}^{3}-160{b_{0}}^{2}{y_{0}}^{5}\right)}{64{b_{0}}^{2}{y_{0}}^{6}}.$
This is a saddle-node bifurcation, where we have two new local equilibria for
${\emph{s}ign}(3a_{1}{y_{0}}+10{b_{0}}^{2}{y_{0}}^{3})\xi_{\pm}<0$. We have no
new equilibrium when
${\emph{s}ign}(3a_{1}{y_{0}}+10{b_{0}}^{2}{y_{0}}^{3})\xi_{\pm}>0$ and
$|\xi_{\pm}|$ is sufficiently small. To obtain the symbolic estimated critical
controller sets (3.5), we consider cubic truncations of $d_{2}(y_{0})=0$ with
respect to $y_{0}.$ Then, $y_{0}$ follows equation (3.3). This confirms our
claim for two local equilibria with a saddle-node singularity. Substituting
them into the truncated equation for $d_{2}(y_{0})=0,$ we derive the estimated
critical controller sets (3.5). ∎

######  Theorem 3.3 (Supercritical and subcritical Hopf bifurcations from
$E_{\pm}$).

We assume that
$|\mu_{0}|={\scalebox{0.7}{$\mathscr{O}$}}(||(\mu_{1},\mu_{2},\mu_{3})||^{4}),$
$|\mu_{3}|={\scalebox{0.7}{$\mathscr{O}$}}(||\mu_{1},\mu_{2}^{2}||),$ and
$a_{1}>0$. Then, approximated Hopf controller sets for degenerate Hopf
singularities associated with $E_{+}$ and $E_{-}$ are defined by $T_{H\pm}$:

$\displaystyle\left\\{\mu|\,2{a_{1}}^{\frac{7}{2}}\mu_{2}-3{a_{1}}^{\frac{5}{2}}\mu_{2}{\mu_{3}}^{2}-4b_{0}{a_{1}}^{\frac{3}{2}}(a_{1}-2b_{0}\mu_{2}-{\mu_{3}}^{2})(\mu_{1}+{\mu_{2}}^{2})\pm\frac{{a_{1}}^{2}(6a_{1}-22b_{0}\mu_{2}-3{\mu_{3}}^{2})\mu_{3}\sqrt{-\mu_{1}-{\mu_{2}}^{2}}}{2}=0\right\\},\quad$
(3.9)

where $\mu=(\mu_{0},\mu_{1},\mu_{2},\mu_{3}).$ Due to the restrictions on
control coefficients, a full Bautin bifurcation does not occur here; see
Theorem 3.7. Yet, one tertiary limit cycle $\mathscr{C}^{1}_{\pm}$ bifurcates
from either of the equilibria $E_{\pm}$ when controller coefficients cross
critical controller sets $T_{H\pm}$ and $b_{0}\eta_{\pm}>0$, respectively.
Here,
$\eta_{\pm}:=\mu_{2}(-\mu_{1}-{\mu_{2}}^{2})^{\frac{-1}{2}}\pm\frac{3}{2}\frac{\mu_{3}}{\sqrt{a_{1}}}+\frac{2b_{0}\sqrt{-\mu_{1}-{\mu_{2}}^{2}}}{a_{1}}.$
The bifurcation is supercritical when $\eta_{\pm}>0$ and subcritical for
$\eta_{\pm}<0.$ The leading terms for the radius and angular velocity of the
bifurcated limit cycles are
$\frac{8\sqrt{2}\sqrt{7a_{1}{b_{0}}^{-1}\mu_{2}-46(\mu_{1}+{\mu_{2}}^{2})}}{7a_{1}\emph{sign}(b_{0})}-\frac{64\sqrt{-\mu_{1}-{\mu_{2}}^{2}}}{7a_{1}}$
and $\sqrt{2}(-\mu_{1}-{\mu_{2}}^{2}),$ respectively.

###### Proof.

To facilitate Hopf bifurcation analysis, we transform differential system
(1.3) into an alternative normal form system
$\dot{\check{x}}=\mu_{0}+\mu_{1}\check{y}+2\mu_{2}\check{x}+{\mu_{2}}^{2}\check{y}+3\mu_{3}\check{x}\check{y}+a_{1}\check{y}^{3}+4b_{0}\check{x}\check{y}^{2}+2b_{0}\mu_{2}\check{y}^{3}+{b_{0}}^{2}\check{y}^{5},$
$\dot{\check{y}}=-\check{x}.$ The estimated secondary equilibria $E_{\pm}$ in
the new coordinate system turns out to be $(\check{x},\check{y})=(0,y_{\pm}),$
where $y_{\pm}$ is given in (3.3). Consider changes of controller coefficients
$\mu_{1}:=-{\nu_{2}}^{2}-{\mu_{2}}^{2}$ and $\mu_{2}:=\nu_{3}\nu_{2}$. Then,
four-degree truncated traces (multiplied with ${\nu_{2}}^{-1}$) of Jacobian
matrices at the equilibria $(\check{x},\check{y})=(0,y_{\pm})$ follow

$\displaystyle 8b_{0}{a_{1}}^{\frac{5}{2}}\nu_{2}\pm
6{a_{1}}^{3}\mu_{3}+4{a_{1}}^{\frac{7}{2}}\nu_{3}\mp
3{a_{1}}^{2}{\mu_{3}}^{3}-16{a_{1}}^{\frac{3}{2}}{b_{0}}^{2}\nu_{3}{\nu_{2}}^{2}$
$\displaystyle\mp
22b_{0}{a_{1}}^{2}\mu_{3}\nu_{2}\nu_{3}-8{\mu_{3}}^{2}b_{0}{a_{1}}^{\frac{3}{2}}\nu_{2}-6{a_{1}}^{\frac{5}{2}}{\mu_{3}}^{2}\nu_{3}=0.$

This gives rise to Hopf controller manifolds for $T_{H\pm}$ in (3.9). For
these Hopf singularities, we consider the quadratic truncated traces of
Jacobian matrices and obtain the variable $\nu_{3}.$ This gives rise to the
introduction of
$\eta_{\pm}:=\nu_{3}\pm\frac{3}{2}\frac{\nu_{4}}{\sqrt{a_{1}}}+\frac{2b_{0}\nu_{2}}{a_{1}}.$
This is the same $\eta_{\pm}$ as in the above in terms of $\mu_{i}.$ Now we
shift the equilibria to the origin via
$\tilde{x}=\check{x},\;\tilde{y}=\check{y}-y_{\pm}.$ Next, we apply linear
transformations $X=\sqrt{2}\sqrt{-\mu_{1}-{\mu_{2}}^{2}}\tilde{y},$
$Y=\tilde{x},$ primary shift of coordinates and a time rescaling
$\tau=\nu_{2}t$ to obtain $\dot{X}=-\sqrt{2}{\nu_{2}}^{2}Y$ and $\dot{Y}$,
where $\dot{Y}$ is given by

$\displaystyle\left(\sqrt{2}{\nu_{2}}^{2}\mp\frac{9\sqrt{2}b_{0}{\nu_{2}}^{3}\mu_{3}}{2a_{1}\sqrt{a_{1}}}-{\frac{7\sqrt{2}{b_{0}}^{2}{\nu_{2}}^{4}}{2{a_{1}}^{2}}}-{\frac{3\sqrt{2}{\nu_{2}}^{2}{\mu_{3}}^{2}}{2a_{1}}}\pm{\frac{2\sqrt{2}\eta{\nu_{2}}^{2}\mu_{3}}{\sqrt{a_{1}}}}\pm{\frac{8b_{0}{\nu_{2}}^{2}Y}{\sqrt{a_{1}}}}\pm{\frac{3\sqrt{2}\eta
b_{0}{\nu_{2}}^{3}}{a_{1}}}+3\mu_{3}\nu_{2}Y\right)X$
$\displaystyle+\left(4b_{0}\nu_{2}Y\mp{\frac{\sqrt{2}{b_{0}}^{2}{\nu_{2}}^{3}}{a_{1}\sqrt{a_{1}}}}\pm{\frac{3\sqrt{2}\eta
b_{0}{\nu_{2}}^{2}}{\sqrt{a_{1}}}}+\sqrt{2}\eta\nu_{2}\mu_{3}-{\frac{\sqrt{2}b_{0}{\nu_{2}}^{2}\mu_{3}}{2a_{1}}}\pm\frac{3\sqrt{2a_{1}}\nu_{2}}{2}\right)X^{2}+h.o.t..$

By a Maple programming (e.g., [22]), an estimated parametric normalized
amplitude equation is given by

$\displaystyle\dot{\rho}=\rho(A_{1}+A_{2}R+A_{3}R^{2})+{\scalebox{0.7}{$\mathscr{O}$}}(\rho^{6}),\;\;A_{1}=\eta_{\pm}\sqrt{-\mu_{1}-{\mu_{2}}^{2}},\;\>A_{2}=-b_{0}\sqrt{-\mu_{1}-{\mu_{2}}^{2}}\pm\frac{9}{16}\mu_{3}\sqrt{a_{1}},$

and $A_{3}=-\frac{7a_{1}b_{0}}{128}.$ Here $R:=\rho^{2}.$ For sufficiently
small values of $\eta,$ the discriminant of $A_{1}+A_{2}R+A_{3}R^{2}$ is
always positive. Since
$|\mu_{3}|={\scalebox{0.7}{$\mathscr{O}$}}(||\mu_{1},\mu_{2}^{2}||)$ and
$\frac{A_{2}}{A_{3}}=\frac{128\sqrt{-{\nu_{3}}^{2}-\nu_{2}}}{-7a_{1}}+\frac{72\mu_{3}}{7b_{0}\sqrt{a_{1}}},$
the sum of its roots is always negative while
$\frac{A_{1}}{A_{3}}=-\frac{128\eta_{\pm}\sqrt{-{\nu_{3}}^{2}-\nu_{2}}}{7a_{1}b_{0}}$
is negative iff $b_{0}\eta_{\pm}>0.$ Indeed, we have a positive root only when
$b_{0}\eta_{\pm}>0$ and otherwise, there is no positive root. These determine
when the system admits a local limit cycle. Despite the degeneracy of Hopf
singularity, controller restrictions limit the bifurcations to at most one
limit cycle from either of $E_{\pm}.$ ∎

(a) Critical controller sets (b) Region I

(c) Region II (d) Region III

(e) Region IV

Figure 1: Controller varieties $\mathscr{B}$ and $\mathscr{H}$ in equations
(3.2) and the numerical steady-state bifurcation diagrams associated with
equation (3.4) and controller coefficient regions I-IV.

(a) $\mu_{3}=0.1,$ $a_{1}=b_{0}=1$ (b) $\mu_{3}=-0.1,$ $a_{1}=b_{0}=1$

(c) $\mu_{3}=0.1,$ $a_{1}=-b_{0}=1$ (d) $\mu_{3}=-0.1,$ $a_{1}=-b_{0}=1$

Figure 2: Estimated critical controller sets for system (1.3) when
$\mu_{0}:=0.$

Figures 2 demonstrate the estimated critical controller sets associated with
system (1.3) when $\mu_{3}:=\pm 0.1$ for
$\mu_{1}={\scalebox{0.7}{$\mathscr{O}$}}(||\mu_{2},\mu_{3}||^{2})$ and
$|\mu_{0}|\ll 1.$ These figures include a pitchfork controller set $T_{P},$
where two equilibria $E_{\pm}$ collide with the origin. Hopf controller sets
for the origin and $E_{\pm}$ are denoted by $T_{H}$, $T_{H+}$ and $T_{H-}$,
according to Proposition 3.1 and Theorem 3.3. Each of the bifurcated limit
cycles disappears when controller coefficients pass through the homoclinic
controller sets $T_{HmC}$ and $T_{HmC\pm}$. More precisely, the limit cycles
$\mathscr{C}^{1}_{\pm}$ (bifurcated from $E_{\pm}$) grow in size and collide
with the origin. These construct homoclinic cycles $\Gamma_{\pm}$; see Figure
3(c). The next theorem deals with deriving the corresponding estimated
controller sets $T_{HmC\pm}$ and homoclinic orbits.

######  Remark 3.4.

Simultaneous collisions of $\mathscr{C}^{1}_{+}$ and $\mathscr{C}^{1}_{-}$
with the origin give rise to a saddle-connection (double homoclinic). This is
the only dynamics possibility for the equivariant cases; see [18]. When
$\mu_{0}={\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|^{2}),$

$\displaystyle
T_{SC}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{2}=\frac{8b_{0}}{5a_{1}}\mu_{1}+\sqrt{-\mu_{1}}{\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|,|\frac{{\mu_{0}}^{2}}{{\mu_{1}}^{3}}|)\right\\}$

is the estimated saddle-connection variety. This is a tuning controller
coefficient manifold between (in the middle of) $T_{HmC+}$ and $T_{HmC-}$; see
[18, Lemma 5.6] and equations (3.10). The relative geometry of these manifolds
determine their validity. For example, when the limit cycles
$\mathscr{C}^{1}_{+}$ and $\mathscr{C}^{1}_{-}$ have already disappeared
through homoclinic bifurcations, the saddle-connection variety is no longer
valid.

An efficient nonlinear time transformation method has been recently developed
and applied for global bifurcation varieties of homoclinic and heteroclinic
varieties of codimension two singularities [2, 35, 1]. This is an efficient
alternative approach to the classical use of Melnikov functions; e.g., see
[29, 32]. Both approaches have been usually applied using one-small scaling
variable. Since all parameters are scaled using one parameter, the approach
typically lead to a one-dimensional transition variety and fits well within a
codimension-two singularity. Transition varieties must have a dimension of
three in order that they would partition the parameter space in four
dimensions. Although the scaling constants play a role in accommodating the
higher dimensional transitions sets (e.g., see [36]), we include three scaling
parameters $\epsilon_{1},\epsilon_{2},\epsilon_{3}.$ We derive an estimation
for controller sets for homoclinic and heteroclinic bifurcations. Our symbolic
estimations are accurate enough for many control engineering applications.
Higher order approximations than our derived formulas are also feasible, but
it is beyond the scope of this paper; e.g., see [2, 36, 35, 1] for highly
accurate one- and two-dimensional transition varieties. Symbolic estimations
for these bifurcations are useful for an efficient management of the nearby
oscillating dynamics.

######  Theorem 3.5 (Homoclinic cycles ${\Gamma}_{\pm}$).

When $a_{1}>0$ and $\mu_{0}={\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|^{2}),$
the bifurcated limit cycles disappear via two distinct quaternary homoclinic
controller sets estimated by

$T_{HmC{{\pm}}}:=\left\\{\mu\big{|}\,\mu_{2}=\frac{8b_{0}}{5a_{1}}\mu_{1}\pm\dfrac{9\sqrt{2}\pi}{32}\mu_{3}\sqrt{-\mu_{1}}\mp\frac{9\sqrt{2}\pi}{32}\frac{\mu_{0}}{\sqrt{-\mu_{1}}}+\sqrt{-\mu_{1}}\,{\scalebox{0.7}{$\mathscr{O}$}}\left(|\mu_{1}|,\left|\frac{{\mu_{0}}^{2}}{{\mu_{1}}^{3}}\right|\right)\right\\},$
(3.10)

for $\mu=(\mu_{0},\mu_{1},\mu_{2},\mu_{3}).$ The leading estimated terms for
the homoclinic cycles $\Gamma_{\pm}$ give rise to an effective criteria for
the magnitude control of the nearby oscillating dynamics. These are given by

$\displaystyle(x(\varphi),y(\varphi))=\left(-\frac{\sin^{2}(\varphi)\cos(\varphi)\sqrt{3-cos(2\varphi)}}{\sqrt{a_{1}}}\mu_{1},\pm\frac{\sqrt{2}}{2}\sqrt{-\mu_{1}}(\cos(2\varphi)-1)\right)+({\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|^{\frac{3}{2}}),{\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|)),$

for $\varphi\in[0,\pi].$

###### Proof.

We apply a nonlinear time transformation method and include multiple scaling
parameters $\epsilon_{i}$ for $i=1,2,3$; see [36, 35, 2]. Namely, we use the
rescaling transformations
$x={\epsilon_{1}}^{2}\tilde{x},y=\epsilon_{1}\tilde{y},$

$\displaystyle
t={\epsilon_{1}}^{-1}\tilde{t},\mu_{0}={\epsilon_{1}}^{3}\left(\gamma_{1}+\gamma_{01}\epsilon_{1}+\gamma_{02}\epsilon_{2}\right),\mu_{1}={\epsilon_{1}}^{2}\left(\gamma_{2}+{\epsilon_{1}}\gamma_{11}+\epsilon_{2}\gamma_{12}+\epsilon_{3}\gamma_{13}\right),$
$\displaystyle\\!\\!\\!\mu_{2}={\epsilon_{1}}^{2}\gamma_{21}+\epsilon_{1}\epsilon_{2}\gamma_{22}+{\epsilon_{1}}\epsilon_{3}\gamma_{23}+\epsilon_{1}{\scalebox{0.7}{$\mathscr{O}$}}(|(\epsilon_{1},\epsilon_{2},\epsilon_{3})|^{2}),\mu_{3}={\epsilon_{1}}\gamma_{31}+\epsilon_{2}\gamma_{32}+\epsilon_{3}\gamma_{33}+{\epsilon_{1}}\epsilon_{2}\gamma_{34}.\;$
(3.11)

These transform the differential system (1.3) into

$\displaystyle\dot{\tilde{x}}$ $\displaystyle=$
$\displaystyle\gamma_{1}\left(\gamma_{2}\epsilon_{1}+\gamma_{02}\epsilon_{2}\right)+\gamma_{2}\left(1+\epsilon_{1}\gamma_{11}+\epsilon_{2}\gamma_{12}+\epsilon_{3}\gamma_{13}\right)\tilde{y}+\left(\epsilon_{1}\gamma_{21}+\epsilon_{2}\gamma_{22}+\epsilon_{3}\gamma_{23}\right)\tilde{x}+a_{1}\tilde{y}^{3}$
(3.12)
$\displaystyle+\left(\epsilon_{1}\gamma_{31}+\epsilon_{2}\gamma_{32}+\epsilon_{3}\gamma_{33}+\epsilon_{1}\epsilon_{2}\gamma_{34}\right)\tilde{x}\tilde{y}+\epsilon_{1}b_{0}\tilde{x}\tilde{y}^{2},$
$\displaystyle\dot{\tilde{y}}$ $\displaystyle=$
$\displaystyle-\tilde{x}+\left(\epsilon_{1}\gamma_{21}+\epsilon_{2}\gamma_{22}+\epsilon_{3}\gamma_{23}\right)\tilde{y}+\left(\epsilon_{1}\gamma_{31}+\epsilon_{2}\gamma_{32}+\epsilon_{3}\gamma_{33}+\epsilon_{1}\epsilon_{2}\gamma_{34}\right)\tilde{y}^{2}+\epsilon_{1}b_{0}\tilde{y}^{3}.$

The unperturbed system, i.e., when
$\epsilon=(\epsilon_{1},\epsilon_{2},\epsilon_{3})=\mathbf{0},$ is a
Hamiltonian system with Hamiltonian
$H=\gamma_{1}\tilde{y}+\frac{1}{2}\tilde{x}^{2}+\frac{1}{2}\gamma_{2}\tilde{y}^{2}+\frac{1}{4}a_{1}\tilde{y}^{4}.$
We further Taylor-expand the new state variables and a time-rescaling
transformation in terms of the scaling parameters $\epsilon_{i}$ for $i=1,2,3$
as

$\displaystyle\tilde{x}(\varphi):=\tilde{x}_{0}(\varphi)+\sum{\epsilon_{j}}^{i}\tilde{x}_{ij}(\varphi),\quad\tilde{y}(\varphi):=\tilde{y}_{0}(\varphi)+\sum{\epsilon_{j}}^{i}\left(p_{ij}\cos(2\varphi)+q_{ij}\right),$
$\displaystyle\tilde{t}=\Phi\tau,\quad\Phi:=\phi_{0}+\sum{\epsilon_{j}}^{i}\phi_{ij},$
(3.13)

where the sum $\sum$ without indices stands for the double sum
$\sum_{i=1}^{\infty}\sum_{j=1}^{3}$ and $\varphi\in[0,\pi].$ Let
$\gamma_{1}:=0,$ $\gamma_{01}:=0,$ $\gamma_{02}:=0,$ $\gamma_{2}:=-1,$
$\gamma_{34}:=1.$ Then, Hamiltonian of the unperturbed system holds a
homoclinic cycle that connects the stable and unstable manifolds of the
origin, i.e., the homoclinic orbit follows $H(\tilde{x},\tilde{y})=0$. When
the rescaling variables $\epsilon_{i}$ for $i=1,2,3$ becomes non-zero, the
homoclinic cycle still holds for a homoclinic variety of codimension-one in
the parameter space. The idea of the nonlinear time transformation method is
to iteratively calculate the homoclinic cycle and homoclinic variety in terms
of powers of $\epsilon_{i}.$ We here only deal with zero and first order
approximations, i.e., $(p_{0},q_{0},x_{0},\phi_{0})$ and
$(p_{1j},q_{1j},x_{1j},\phi_{1,j})$ for $j=1,2,3$. We remark that there is
only a homoclinic cycle for system (3.12). However, this will turn out to be
two homoclinic cycles $\Gamma_{\pm}$ for (1.3), depending on the sign of
$\epsilon_{1}$ in (3.18). The zero order approximation is given by
$(\tilde{x}_{0}(\varphi),\tilde{y}_{0}(\varphi)),$ where we assume that

$\displaystyle\tilde{y}_{0}:=p_{0}\cos(2\varphi)+q_{0}\quad\hbox{ and
}\quad\tilde{x}_{0}(0)=\tilde{x}_{0}(\frac{\pi}{2})=0.$ (3.14)

Hence,
$(\tilde{y}_{0}(0),\tilde{y}_{0}(\frac{\pi}{2}))=(p_{0}+q_{0},q_{0}-p_{0}).$
Since Hamiltonian is constant over the homoclinic cycle, we have
$H(\tilde{x}_{0}(\frac{\pi}{2}),\tilde{y}_{0}(\frac{\pi}{2}))=H(0,p_{0}+q_{0})$.
Furthermore, $\frac{\partial H}{\partial\tilde{y}}(0,p_{0}+q_{0})=0$ due to
the fact that $(\tilde{x}_{0}(0),\tilde{y}_{0}(0))$ is an equilibrium for the
unperturbed Hamiltonian system. These equations give rise to

$\displaystyle p_{0}=\frac{\sqrt{2}}{2\sqrt{a_{1}}},\quad
q_{0}=-\frac{\sqrt{2}}{2\sqrt{a_{1}}},\qquad\tilde{y}_{0}=\frac{\sqrt{2}}{2\sqrt{a_{1}}}\cos(2\varphi)-\frac{\sqrt{2}}{2\sqrt{a_{1}}},$
$\displaystyle\tilde{x}_{0}=\pm\frac{\sin^{2}(\varphi)\cos(\varphi)\sqrt{3-cos(2\varphi)}}{\sqrt{a_{1}}},\quad\hbox{
and }\quad\phi_{0}(\varphi):=-\frac{\tilde{x}_{0}}{{\tilde{y}_{0}}^{\prime}}.$

Let $q_{1j}:=0$ for $j=1,2,3.$ Then, $\tilde{y}_{11}=p_{11}\cos(2\varphi),$
$\tilde{y}_{12}=p_{12}\cos(2\varphi),$ $\tilde{y}_{13}=p_{13}\cos(2\varphi)$
and the first-order approximation follows

$\displaystyle\tilde{x}=\tilde{x}_{0}+\epsilon_{1}\tilde{x}_{11}+\epsilon_{2}\tilde{x}_{12}+\epsilon_{3}\tilde{x}_{13},\;\tilde{y}=\tilde{y}_{0}+\epsilon_{1}p_{11}\cos(2\varphi)+\epsilon_{2}p_{12}\cos(2\varphi)+\epsilon_{3}p_{13}\cos(2\varphi).\quad$
(3.15)

Next, the terms of the first-order in terms of $\epsilon_{i}$ for $i=1,2,3$ in
$\phi\dot{x}$ give rise to

$\displaystyle\\!\\!\\!\\!\\!\phi_{0}x_{11}^{\prime}+p_{11}\cos{(2\varphi)}-x_{0}\gamma_{21}-x_{0}y_{0}\gamma_{31}-y_{0}\gamma_{11}-\gamma_{01}-3a_{1}{y_{0}}^{2}p_{11}\cos{(2\varphi)}+\phi_{11}x_{0}^{\prime}-b_{0}x_{0}{y_{0}}^{2}=0,$
$\displaystyle\phi_{0}x_{1i}^{\prime}+p_{1i}\cos{(2\varphi)}-x_{0}\gamma_{3(i+1)}-x_{0}y_{0}\gamma_{4(i+1)}-y_{0}\gamma_{2(i+1)}$
$\displaystyle-\gamma_{1(i+1)}-3a_{1}{y_{0}}^{2}p_{1i}\cos{(2\varphi)}+\phi_{1i}x_{0}^{\prime}=0,\quad$
(3.16)

see also [35, Equation 2.22a and 2.22b]. Now consider the first-order
$\epsilon_{i}$-terms in $\phi\dot{y}$ along with equations (3.16). By
eliminating $\phi_{1j}$-terms from these equations, an integrating factor and
an integration, similar to the proof of [35, Equation 2.30], we derive

$\displaystyle\int^{\varphi}_{0}y_{0}^{\prime}\big{(}y_{0}\gamma_{11}-p_{12}\cos{(2\varphi)}+x_{0}\gamma_{21}+y_{0}x_{0}\gamma_{31}+b_{0}x_{0}{y_{0}}^{2}+\gamma_{01}+3a_{1}{y_{0}}^{2}p_{12}\cos{(2\varphi)}\big{)}d\varphi$
$\displaystyle+x_{0}x_{11}+y_{11}g(y_{0})+\int^{\varphi}_{0}x_{0}^{\prime}\left({y_{0}}^{2}\gamma_{31}+\gamma_{21}y_{0}+b_{0}{y_{0}}^{3}\right)d\varphi=0,$
(3.17)
$\displaystyle\int^{\varphi}_{0}y_{0}^{\prime}\big{(}y_{0}\gamma_{2(i+1)}-p_{1i}\cos{(2\varphi)}+x_{0}\gamma_{3(i+1)}+y_{0}x_{0}\gamma_{4(i+1)}+\gamma_{1(i+1)}+3a_{1}{y_{0}}^{2}p_{1i}\cos(2\varphi)\big{)}d\varphi$
$\displaystyle\qquad\qquad\;\;+x_{0}x_{1i}+y_{1i}g(y_{0})+\int^{\varphi}_{0}x_{0}^{\prime}\left({y_{0}}^{2}\gamma_{4(i+1)}+\gamma_{3(i+1)}y_{0}\right)d\varphi=0,\hbox{
for }i=2,3.$

Evaluating equation (3.16) at $\varphi=\pi$ and (3.17) at $\varphi=\pi,\pi/2$,
we obtain nine number of linear equations. These give rise to the scaling
parameters

$\displaystyle\gamma_{21}=-\frac{8b_{0}}{5a_{1}}+\frac{9\sqrt{2}\pi}{32}\gamma_{31},\quad\gamma_{22}=\frac{9\sqrt{2}\pi}{32}\gamma_{32},\quad\gamma_{23}=\frac{9\sqrt{2}\pi}{32}\gamma_{33},$
(3.18)
$\displaystyle\gamma_{11}=0,\quad\gamma_{12}=0,\quad\gamma_{13}=0,\quad\epsilon_{1}:=\pm\sqrt{-\mu_{1}}.$

Finally, we substitute these into the equation for $\mu_{2}$ in (3.11) and
derive transition sets $T_{HmC{{\pm}}}$ given in equation (3.10). ∎

Theorem 3.3 implies that control coefficients
$(\mu_{0},\mu_{1},\mu_{2},\mu_{3})$ for
$|\mu_{0}|={\scalebox{0.7}{$\mathscr{O}$}}(||(\mu_{1},\mu_{2},\mu_{3})||^{4}),$
$|\mu_{3}|={\scalebox{0.7}{$\mathscr{O}$}}(||\mu_{1},\mu_{2}^{2}||)$ are not
enough for fully unfolding a Bautin bifurcation around $E_{\pm}.$ Now we show
that system (1.3) undergoes full Bautin bifurcation scenarios (in particular,
bifurcations of two limit cycles and saddle-node bifurcation of limit cycles)
when these restrictions on control coefficients are removed. For the
distinction of different bifurcation scenarios, we will instead denote the
equilibria with $E^{*}_{\pm}$ in Lemma 3.6, Theorem 3.7 and Remark 3.8.

######  Lemma 3.6 (Critical controller sets and normalized amplitude equation
for generalised Hopf singularity).

Let $9{\mu_{3}}^{2}\geq 32b_{0}\mu_{2},$
$\delta:=\sqrt{9{\mu_{3}}^{2}-32b_{0}\mu_{2}},$ and $\zeta_{\pm}$ be given by

$\displaystyle\zeta_{\pm}=\mu_{0}-\frac{\mu_{3}(7{\mu_{2}}^{2}+12\mu_{1})}{32b_{0}}+\frac{9\mu_{2}\mu_{3}(3{\mu_{3}}^{2}+16a_{1})}{256{b_{0}}^{2}}-\frac{27{\mu_{3}}^{3}({\mu_{3}}^{2}+16a_{1})}{2048{b_{0}}^{3}}$
(3.19)
$\displaystyle\qquad\quad\pm\frac{\delta\left(9{\mu_{3}}^{4}-56b_{0}\mu_{2}{\mu_{3}}^{2}+64{b_{0}}^{2}{\mu_{2}}^{2}+144a_{1}{\mu_{3}}^{2}-128a_{1}b_{0}\mu_{2}+256{b_{0}}^{2}\mu_{1}\right)}{2048{b_{0}}^{3}}.$

Further, let
$\kappa:=432a_{1}{\mu_{3}}^{2}-3\mu_{3}\delta\left(48a_{1}+16b_{0}\mu_{2}-3{\mu_{3}}^{2}\right)-768a_{1}b_{0}\mu_{2}+512{b_{0}}^{2}\mu_{1}+192b_{0}\mu_{2}{\mu_{3}}^{2}-384{b_{0}}^{2}{\mu_{2}}^{2}-27{\mu_{3}}^{4}$
and $\zeta={\scalebox{0.7}{$\mathscr{O}$}}(\kappa).$ Then,

  1. a.

There are two generalized Hopf singularities for $E^{*}_{\pm}$ at
$\zeta_{\pm}=0.$

  2. b.

These generalized Hopf singularities are determined by the cubic jet of the
system (1.3).

  3. c.

The normalized amplitude equation of the cubic truncated system is given by

$\displaystyle\dot{\rho}_{\pm}=\pm
256\,\delta\zeta_{\pm}\rho+\frac{\kappa}{2{b_{0}}}\rho^{3}-64a_{1}{b_{0}}\rho^{5}+{\scalebox{0.7}{$\mathscr{O}$}}(\rho^{7}).$
(3.20)

###### Proof.

Since we are dealing with a Bautin bifurcation, we initially consider the
truncated normal form system (1.3) up to degree five, that is,

$\displaystyle\dot{x}=\mu_{0}+\mu_{1}y+\mu_{2}x+\mu_{3}xy+a_{0}y^{3}+b_{0}xy^{2}+b_{1}xy^{4},\,\dot{y}=-x+\mu_{2}y+\mu_{3}y^{2}+b_{0}y^{3}+b_{1}y^{5}.$
(3.21)

We can show that the corresponding second Lyapunov coefficients is estimated
by

$\displaystyle\frac{3}{256}b_{0}b_{1}\kappa^{2}-64a_{0}b_{0}-15a_{0}b_{0}b_{1}\delta^{2}+\frac{53}{16}b_{0}\delta^{2}.$

For all values of $b_{1}$ and sufficiently small choices for the parameters,
we can ensure that the second Lyapunov coefficient is always non-zero. In
other words, Bautin bifurcation here is completely determined by the cubic
terms of the differential system (1.3). We can thus truncate the normal form
system (1.3) at degree three, i.e., let $b_{1}:=0$. Now recall $d_{1}$ and
$d_{2}$ from the first column of Routh table in equations (3.6). We solve
$\dot{y}=0$ for $x$ and substitute it into $\dot{x}=0$ to obtain

$\displaystyle\mu_{0}+\mu_{1}y+{\mu_{2}}^{2}y+2\mu_{2}\mu_{3}y^{2}+2b_{0}\mu_{2}y^{3}+{\mu_{3}}^{2}y^{3}+2b_{0}\mu_{3}y^{4}+a_{1}y^{3}+{b_{0}}^{2}y^{5}=0.$
(3.22)

Hopf bifurcation occurs when $d_{1}=0$ and $d_{2}>0.$ Thus, there are two
local Hopf singularities at

$\displaystyle E^{*}_{\pm}:=\left(x^{*}_{\pm},y^{*}_{\pm}\right),\quad\hbox{
for }\;y^{*}_{\pm}:=\frac{-3\mu_{3}\pm\delta}{8b_{0}}\;\hbox{ and
}\;x^{*}_{\pm}:=\mu_{2}y^{*}_{\pm}+\mu_{3}{y^{*}_{\pm}}^{2}+b_{0}{y^{*}_{\pm}}^{3}.$

Their associated controller sets follow (3.19). Using shift of coordinates
$(x,y)-(x^{*}_{\pm},y^{*}_{\pm})$ on equation (1.3) and transforming the
linear part into Jordan canonical form, we obtain
$\dot{x}=-\frac{1}{32}\sqrt{\kappa}y\pm\frac{3}{8}\delta
x^{2}-\frac{1}{8}\mu_{3}x^{2}+b_{0}x^{3}$ and

$\displaystyle\dot{y}=\frac{1}{32}\sqrt{\kappa}x+\frac{64a_{1}b_{0}x^{3}-64b_{0}\eta^{\pm}+9{\mu_{3}}^{3}x^{2}\pm
8b_{0}\delta\mu_{2}x^{2}\mp
3\delta{\mu_{3}}^{2}x^{2}-72a_{1}\mu_{3}x^{2}-40x^{2}b_{0}\mu_{2}\mu_{3}-24a_{1}\delta
x^{2}}{2b_{0}\sqrt{\kappa}}$
$\displaystyle+\frac{1}{4}xy\left(4b_{0}x+\mu_{3}\pm\delta\right).$

A computer programming (e.g., [16]) gives rise to the normalized equation
(3.20). ∎

######  Theorem 3.7 (Bautin bifurcations from $E^{*}_{\pm}$).

Let $9{\mu_{3}}^{2}\geq 32b_{0}\mu_{2}$ and
$\zeta_{\pm}={\scalebox{0.7}{$\mathscr{O}$}}(\kappa).$ Then for $a_{1}<0,$
there are one supercritical and one subcritical Hopf controller sets estimated
by

$\displaystyle
T^{Sup}_{H\pm}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})\,|\,\zeta_{+}=0,b_{0}<0\right\\}\;\hbox{
and }$
$\displaystyle\;T^{Sub}_{H\pm}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})\,|\,\zeta_{-}=0,b_{0}>0\right\\}.$
(3.23)

When controller coefficients cross $T^{Sup_{1}}_{H+}$ given by (3.19) and
$\zeta_{+}={\scalebox{0.7}{$\mathscr{O}$}}(\kappa),$ one stable limit cycle
$\mathscr{C}^{1}_{+}$ bifurcates from $E^{*}_{+}.$ As for $T^{Sub_{1}}_{H-}$
when $\zeta_{-}={\scalebox{0.7}{$\mathscr{O}$}}(\kappa),$ the bifurcation
causes an unstable local limit cycle $\mathscr{C}^{1}_{-}$ encircling
$E^{*}_{-}.$ Both of these limit cycles are considered as tertiary limit
cycles. Two simultaneous limit cycles surrounding $E^{*}_{+}$ (or $E^{*}_{-}$)
do not appear when $a_{1}<0$ and naturally, saddle-nodes of limit cycles does
not occur in this case. For $a_{1}>0$ and
$\zeta_{\pm}={\scalebox{0.7}{$\mathscr{O}$}}(\kappa),$ two subcritical and
supercritical Hopf controller sets occur through the estimated manifolds

$\displaystyle
T^{Sub_{1}}_{H\pm}=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})\big{|}\,26214a_{1}\delta\zeta_{\pm}-b_{0}\kappa^{2}=0,\;\pm
b_{0}\zeta_{\pm}>0,b_{0}<0,\mbox{ and }\,a_{1}>0\right\\},$ (3.24)
$\displaystyle
T^{Sup_{1}}_{H\pm}=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})\big{|}\,26214a_{1}\delta\zeta_{\pm}-b_{0}\kappa^{2}=0,\;\pm
b_{0}\zeta_{\pm}>0,b_{0}>0,\mbox{ and }\,a_{1}>0\right\\},$

and

$\displaystyle
T^{Sup_{2}}_{H\pm}=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})\big{|}\,26214a_{1}\delta\zeta_{\pm}-b_{0}\kappa^{2}>0,\;\zeta_{-}=0,b_{0}<0,\mbox{
and }\,a_{1}>0\right\\},$ (3.25) $\displaystyle
T^{Sub_{2}}_{H\pm}=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})\big{|}\,26214a_{1}\delta\zeta_{\pm}-b_{0}\kappa^{2}>0,\;\zeta_{+}=0,b_{0}>0,\mbox{
and }\,a_{1}>0\right\\}.$

When controller coefficients are close to $T^{Sub_{1}}_{H+}$
($T^{Sup_{1}}_{H-}$) and $a_{1}b_{0}\zeta_{+}>0$ ($a_{1}b_{0}\zeta_{-}<0$), we
have only one tertiary limit cycle $\mathscr{C}^{1}_{+}$
($\mathscr{C}^{1}_{-}$) encircling $E^{*}_{+}$ ($E^{*}_{-}$, respectively).
However, a second small limit cycle $\mathscr{C}^{2}_{+}$
($\mathscr{C}^{2}_{-}$) bifurcates from $E^{*}_{+}$ ($E^{*}_{-}$) as soon as
control coefficients cross $T^{Sub_{2}}_{H+}$ ($T^{Sup_{2}}_{H-}$) and
$a_{1}b_{0}\zeta_{+}$ ($a_{1}b_{0}\zeta_{-}$) becomes negative (positive).
Here, we have two pairs of limit cycles ($\mathscr{C}^{1}_{\pm},$
$\mathscr{C}^{2}_{\pm}$) surrounding $E^{*}_{\pm}$, where
$\mathscr{C}^{2}_{\pm}$ lives inside $\mathscr{C}^{1}_{\pm}$. Two more
estimated critical controller sets follow

$T^{\pm}_{SNL}:=\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})\big{|}\,26214a_{1}\delta\zeta_{\pm}-b_{0}\kappa^{2}=0,\pm
b_{0}\zeta_{\pm}<0,a_{1}>0\\},$ (3.26)

where the two limit cycles ($\mathscr{C}^{1}_{\pm},$ $\mathscr{C}^{2}_{\pm}$)
coalesce and disappear as a saddle-node bifurcation of limit cycles.

###### Proof.

Recall the 3-jet normal form amplitude equation (3.20). Let

$\displaystyle p_{\pm}(\rho):=A\rho^{2}+B\rho+C_{\pm},\;\hbox{ where
}\;A:=-64a_{1}{b_{0}},\qquad B:=\frac{1}{2}b_{0}\kappa,\qquad C_{\pm}:=\pm
256\delta\zeta_{\pm}.$

Positive roots of $p_{\pm}$ correspond with the limit cycles bifurcated from
$E^{*}_{\pm}.$ We first remark that $\delta$ is always non-negative. Since
$\kappa>0,$ $\frac{-B}{A}=\frac{\kappa}{128a_{1}}$ is always negative for
$a_{1}<0$ and at most one limit cycle can bifurcate. In the case of $a_{1}<0$,
we have one limit cycle for
$\frac{C_{\pm}}{A}=\mp\frac{4\delta\zeta_{\pm}}{a_{1}b_{0}}<0,$ and no
positive root for ${\emph{sign}}(\frac{C_{\pm}}{A})={\emph{sign}}(\pm
b_{0}\zeta_{\pm})>0.$ When the limit cycle exists, i.e., $\pm
b_{0}\zeta_{\pm}<0,$ the limit cycle is asymptotically stable for
$\pm\zeta_{\pm}>0.$ Therefore, $\zeta_{\pm}=0$ is a critical controller set
for the appearance of a limit cycle and it is supercritical when $b_{0}<0.$
This critical controller set is subcritical for positive values of $b_{0}$.
These arguments justify $T^{Sup}_{H\pm}$ and $T^{Sub}_{H\pm}$ in (3.7).

Let $a_{1}>0.$ Hence, $\frac{-B}{A}>0.$ Assume that
$B^{2}-4AC_{\pm}=\pm\frac{b_{0}\left(102\times
257a_{1}\delta\zeta_{\pm}-b_{0}\kappa^{2}\right)}{4}>0.$ When
$\frac{C_{\pm}}{A}<0,$ the polynomial $p_{\pm}$ has exactly one positive root
while for $\frac{C_{\pm}}{A}>0,$ we have two positive roots for $p_{\pm}.$
Therefore, ${\emph{sign}}(\frac{C_{\pm}}{A})={\emph{sign}}(\mp
b_{0}\zeta_{\pm})<0$ and $B^{2}-4AC_{\pm}=0$ is a critical controller manifold
where one limit cycle bifurcates from $E^{*}_{\pm}.$ The bifurcated limit
cycle is asymptotically stable when $C_{\pm}>0$; i.e., $b_{0}>0.$ Thus, $\pm
b_{0}\zeta_{\pm}>0$ and $B^{2}-4AC_{\pm}=0$ gives rise to a supercritical Hopf
controller set $T^{Sup_{1}}_{H\pm}$ in (3.24), where a small limit cycle
bifurcates from $E^{*}_{\pm}.$ Furthermore, $\frac{C_{\pm}}{A}=0$ and
$B^{2}-4AC_{\pm}>0$ is another critical controller set where one small limit
cycles is born in the interior of an already existed larger limit cycle. The
new small bifurcated limit cycle is asymptotically stable when $\mp
b_{0}\zeta_{\pm}>0$ and $\pm\zeta_{\pm}>0.$ This is equivalent with $b_{0}<0$.
These conditions determine the supercritical controller set
$T^{Sup_{2}}_{H\pm}$ in equation (3.25). Thus, the argument for subcritical
controller manifold $T^{Sub_{2}}_{H\pm}$ for $b_{0}>0$ is similar. For
$\frac{C_{\pm}}{A}>0,$ we have two different cases:

  1. 1.

We have two positive roots for $B^{2}-4AC_{\pm}>0.$

  2. 2.

There is no limit cycle for $B^{2}-4AC_{\pm}<0.$

Hence, when $\pm b_{0}\zeta_{\pm}<0$ and $a_{1}>0$ hold, $B^{2}-4AC_{\pm}=0$
is a saddle-node controller set $T^{\pm}_{SNL}$ of limit cycles in (3.26),
where two limit cycles coalesce and disappear. ∎

(a) $a_{1}=b_{0}=1,\mu_{1}=-0.02,\mu_{2}=-0.06$ (b)
$a_{1}=b_{0}=1,\mu_{0}=0.001,\mu_{3}=0.1$

(c) $a_{1}=b_{0}=1,$ $\mu_{0}=.001,$ $\mu_{1}=-.1$ $\mu_{2}=-.122,$
$\mu_{3}=.1$ (d) $a_{1}=-b_{0}=-1,$ $\mu_{0}=0,$ $\mu_{1}=.01,$
$\mu_{2}=.00394,$ $\mu_{3}=.01$

Figure 3: Estimated controller sets 3(a)-3(b) for (1.3). Estimated
$\Gamma_{-}$ in 3(c) and heteroclinic $\Lambda$ in 3(d).

######  Remark 3.8 (Basin of attraction for stabilization approach via
bifurcated stable limit cycles).

Bautin bifurcation described in Theorem 3.7 includes supercritical and
subcritical bifurcations of tertiary limit cycles $\mathscr{C}^{1}_{\pm}$ and
$\mathscr{C}^{2}_{\pm}.$ They can be used to stabilize the system when the
limit cycle is stable. However, it is important to notice about their basin of
attractions. When $\mathscr{C}^{2}_{\pm}$ is stable, the basin of attraction
is the interior of $\mathscr{C}^{1}_{\pm}\setminus\\{E^{*}_{\pm}\\}$. For
example, Figure 4(b) illustrates the stable limit cycle $\mathscr{C}^{2}_{+}.$
Its basin of attraction is the interior of $\mathscr{C}^{1}_{+}$ (except
$E^{*}_{+}$). The basin of attraction for the stable cases of
$\mathscr{C}^{1}_{\pm}$ includes the region encircled by
$\mathscr{C}^{1}_{\pm}$ and $\mathscr{C}^{2}_{\pm}.$ However, the complete
description for basin of attraction for the exterior of
$\mathscr{C}^{1}_{\pm}$ depends on the dynamics of the system. For instance,
Figure 6(d) demonstrates stable limit cycle $\mathscr{C}^{1}_{+}$ living
inside unstable limit cycle $\mathscr{C}_{0}$. The basin of attraction for
$\mathscr{C}^{1}_{+}$ consists of the region between the stable manifolds (red
and blue curves in Figure 6(d)) of the origin (excluding the equilibria). This
region includes the interior of $\mathscr{C}^{1}_{+}.$

The limit cycle $\mathscr{C}_{0}$ grows in size to collide with either of the
secondary equilibria as controller coefficients go away from the corresponding
Hopf controller set (3.1), i.e., $\frac{\mu_{2}}{b_{0}}$ decreases. The limit
cycle collides with either $E_{+}$ or $E_{-}$ giving rise to homoclinic cycles
$\Lambda_{\pm}$ or simultaneously collides with both of them. The latter leads
to a heteroclinic cycle $\Lambda$; see Figure 3(d). The following two theorems
deal with heteroclinic and homoclinic bifurcations. The limit cycle
$\mathscr{C}_{0}$ may alternatively collide with the origin and disappear
through a homoclinic bifurcation; e.g., see Figure 6(e) where limit cycle
$\mathscr{C}_{0}$ collides with the saddle and disappear as in Figure 6(f).
Estimated controller set is then given by
$T_{HmC}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{2}=\frac{8b_{0}}{5a_{1}}\mu_{1}\right\\}.$

######  Theorem 3.9 (Heteroclinic cycle $\Lambda$ when $a_{1}<0$).

Let $|\mu_{0}|={\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|^{2})$ and $a_{1}<0.$
Then, there is a heteroclinic cycle $\Lambda.$ This connects the equilibrium
$E_{+}$ with the saddle $E_{-}$. The corresponding heteroclinic bifurcation
occurs at the heteroclinic controller manifold approximated by

$T_{HtC}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{2}=\frac{2b_{0}}{5a_{1}}\mu_{1}+\frac{9}{16}\mu_{3}\sqrt{\mu_{1}}-\frac{9}{16}\frac{\mu_{0}}{\sqrt{\mu_{1}}}+{\scalebox{0.7}{$\mathscr{O}$}}(||(\mu_{1},\mu_{3})||^{\frac{3}{2}})\right\\}.$
(3.27)

The most leading estimated terms for $\Lambda$ are

$\displaystyle(x,y)=\left(\mu_{1}\frac{\sqrt{2}}{2}\sin(2\varphi)\sqrt{a_{1}\cos^{2}(2\varphi)+2+a_{1}},\frac{\sqrt{\mu_{1}}}{\sqrt{-a_{1}}}\cos(2\varphi)\right)+({\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|^{\frac{3}{2}}),{\scalebox{0.7}{$\mathscr{O}$}}(|\mu_{1}|)).$

###### Proof.

Here, we use the rescaling transformations (3.11) and (3.13) when
$\gamma_{1}:=0,$ $\gamma_{02}:=0,$ $\gamma_{2}:=1,$
$\gamma_{11}:=\gamma_{12}:=\gamma_{13}:=0,$ and $\gamma_{34}:=1.$ The
unperturbed heteroclinic orbit connects the two saddles
$\big{(}0,\pm\frac{1}{\sqrt{-a_{1}}}\big{)}.$ We assume that equations (3.14)
hold. Similar to [35, Equation 2.11], we have

$\displaystyle
y_{0}(0)=-\frac{1}{\sqrt{-a_{1}}}=p_{0}+q_{0},y_{0}(\frac{\pi}{2})=\frac{1}{\sqrt{-a_{1}}}=q_{0}-p_{0},\hbox{
and thus, }$
$\displaystyle\tilde{y}_{0}=p_{0}\cos(2\varphi)+q_{0}=\frac{1}{\sqrt{-a_{1}}}\cos(2\varphi),q_{0}=0.$

We compute the unperturbed heteroclinic orbit via
$H(\tilde{x}_{0},\tilde{y}_{0})=H(0,\pm(-a_{1})^{\frac{-1}{2}})$ where
$H(\tilde{x},\tilde{y})=\frac{1}{2}\tilde{x}^{2}-\frac{1}{2}\tilde{y}^{2}+\frac{1}{4}a_{1}\tilde{y}^{4}.$
Therefore,
$\tilde{x}_{0}=\frac{\sqrt{2}}{2}\sin(2\varphi)\sqrt{a_{1}\cos^{2}(2\varphi)+2+a_{1}}.$
The first-order terms for $i=1,2,3$ in $\phi\dot{x}$ follow equations (3.16)
and equations (3.17) hold. We need the first-order terms in $\phi\dot{y}$
given by (see [35, Equations 2.22a and 2.22b])

$\displaystyle\phi_{0}y_{11}^{\prime}-\gamma_{21}y_{0}-{y_{0}}^{2}\gamma_{31}-b_{0}{y_{0}}^{3}+\phi_{11}y_{0}^{\prime}=0,\phi_{0}y_{1i}^{\prime}-\gamma_{3(i+1)}y_{0}-{y_{0}}^{2}\gamma_{4(i+1)}+\phi_{1i}y_{0}^{\prime}=0,$
(3.28)

for $i=2,3.$ We evaluate equations (3.16) and (3.28) at
$\varphi=0,\frac{\pi}{2},$ while equations (3.17) are computed at
$\varphi=\pi/2.$ These lead to fifteen linear equations and
$\gamma_{21}=\frac{2b_{0}}{5a_{1}}+\frac{9}{16}\gamma_{31},$
$\gamma_{32}=\frac{16}{9}\gamma_{22},$ and
$\gamma_{33}=\frac{16}{9}\gamma_{23}.$ Thus, transition varieties (3.27) are
derived by substitution of these values into the rescaling transformation for
$\mu_{2}.$ ∎

(a) There is only the source equilibrium $E^{*}_{+}.$ (b) Stable limit cycle
$\mathscr{C}^{2}_{+},$ unstable limit cycle $\mathscr{C}^{1}_{+},$ and a
source equilibrium $E^{*}_{+}$

(c) There is an unstable cycle $\mathscr{C}^{1}_{+}$ and stable equilibrium
$E^{*}_{+}$. (d) Unstable limit cycle $\mathscr{C}^{1}_{+}$, spiral sinks
$E^{*}_{\pm},$ and the primary saddle.

(e) An unstable large limit cycle $\mathscr{C}^{1}_{+},$ a stable limit cycle
$\mathscr{C}^{2}_{-},$ primary saddle, sink $E^{*}_{+}$ and source
$E^{*}_{-}.$ (f) Two limit cycles $\mathscr{C}^{1}_{+}$ and
$\mathscr{C}^{2}_{-}$ surround unstable $E^{*}_{-}.$ Here,
$\mathscr{C}^{1}_{+}$ plays the role of $\mathscr{C}^{1}_{-}$ due to
disappearance of $E^{*}_{+}.$

(g) Asymptotically unstable equilibrium $E^{*}_{-}.$ (h) Spiral sink
$E^{*}_{-}$ and unstable limit cycle $\mathscr{C}^{1}_{+}.$ Here, $E^{*}_{+}$
disappeared at $T^{+}_{SN}$ and $\mathscr{C}^{1}_{+}$ plays the role of
$\mathscr{C}^{1}_{-}.$

Figure 4: Numerical phase portraits 4(a)-4(g) are associated with regions
$a-h$ in 3(a) and (1.3).

######  Theorem 3.10 (Homoclinic controller manifolds for $\Lambda_{\pm}$).

Let $a_{1}<0.$ There are two homoclinic cycles $\Lambda_{\pm}$ connecting the
stable and unstable manifolds of $E_{\pm}$ at homoclinic controller manifolds
estimated by

$\displaystyle
T^{\pm}_{HmC}:=\left\\{(\mu_{0},\mu_{1},\mu_{2},\mu_{3})|\,\mu_{1}=\frac{10^{\frac{2}{3}}}{\left(-a_{1}\right)^{\frac{5}{2}}}{\mu_{0}}^{\frac{2}{3}}-\frac{49.19204541}{\left(-a_{1}\right)^{\frac{5}{2}}}b_{0}\mu_{0}-\frac{8.203865604}{10^{\frac{-2}{3}}\left(-a_{1}\right)^{\frac{11}{6}}}\mu_{2}{\mu_{0}}^{\frac{1}{3}}+\frac{4.355675048}{10^{\frac{-2}{3}}\left(-a_{1}\right)^{3}}\mu_{3}{\mu_{0}}^{\frac{2}{3}}\right\\}.$
(3.29)

The homoclinic $\Lambda_{+}$ occurs when $\mu_{0}>0$ while $\Lambda_{-}$
corresponds with negative values of $\mu_{0}$. The leading estimated terms for
$(x,y)$-coordinates of the homoclinic cycles $\Lambda_{\pm}$ are

$\displaystyle\mp 0.7157063998\cos(2\varphi)\mp 0.2299428741\qquad\qquad\hbox{
and }$ $\displaystyle\mp
0.3622053022\sqrt{2}\sin(\varphi)\sin(2\varphi)\sqrt{cos({2\varphi})+2.28512403471548},\quad\hbox{
for }\varphi\in[0,\pi],$

respectively. This is useful for the management of its nearby oscillating
dynamics.

###### Proof.

We first use transformations $x=\left(-a_{1}\right)^{\frac{3}{2}}\hat{x},$
$y=\left(-a_{1}\right)^{\frac{1}{2}}y,$ and time rescaling
$t=-\frac{1}{a_{1}}\tau$ to change the coefficient $a_{1}$ to $-1$. Then, we
have

$\displaystyle\dot{\hat{x}}=\left(-a_{1}\right)^{\frac{-5}{2}}\mu_{0}+\left(-a_{1}\right)^{-2}\mu_{1}\hat{y}+\left(-a_{1}\right)^{-1}\mu_{2}\hat{x}-\hat{y}^{3}+\left(-a_{1}\right)^{\frac{-1}{2}}\mu_{3}\hat{x}\hat{y}+b_{0}\hat{x}\hat{y}^{2},$
(3.30)
$\displaystyle\dot{\hat{y}}=-\hat{x}+\left(-a_{1}\right)^{-1}\mu_{2}\hat{y}+\left(-a_{1}\right)^{\frac{-1}{2}}\mu_{3}\hat{y}^{2}+b_{0}\hat{y}^{3}.$

Let $\mu_{0}^{\ast}:=\left(-a_{1}\right)^{\frac{-5}{2}}\mu_{0},$
$\mu_{1}^{\ast}:=\left(-a_{1}\right)^{-2}\mu_{1},$
$\mu_{2}^{\ast}:=\left(-a_{1}\right)^{-1}\mu_{2},$
$\mu_{3}^{\ast}:=\left(-a_{1}\right)^{\frac{-1}{2}}\mu_{3}.$ Next, we replace
$\mu_{i}^{\ast}$ with $\mu_{i}$ and $\hat{y}$ with $\tilde{y}$ for simplicity.
Now apply the rescaling transformations (3.11) and expansion (3.13) when

$\gamma_{1}=\pm
0.1,\gamma_{01}=\gamma_{02}=0,\gamma_{2}=-1,\gamma_{11}=\gamma_{13}=0,\gamma_{12}=1,\gamma_{21}=\gamma_{22}=0,\gamma_{34}=0.$
(3.31)

Recall that the unperturbed system is Hamiltonian and it holds a homoclinic
cycle $\Lambda_{+}$ for $\gamma_{1}>0.$ This connect the stable and unstable
manifolds of the saddle $E_{+}.$ The homoclinic cycle $\Lambda_{-}$ happens
for $\gamma_{1}<0$ and corresponds with $E_{-}$. Following the proof of
Theorem 3.5, we apply equations (3.14),
$(\tilde{y}_{0}(0),\tilde{y}_{0}(\frac{\pi}{2}))=(p_{0}+q_{0},q_{0}-p_{0}),$
$H(\tilde{x}_{0}(\frac{\pi}{2}),\tilde{y}_{0}(\frac{\pi}{2}))=H(0,p_{0}+q_{0}),$
and $\frac{\partial H}{\partial\tilde{y}}(0,p_{0}+q_{0})=0$ to obtain

$\displaystyle\tilde{y}_{0}=\mp 0.7157063998\cos(2\varphi)\mp
0.2299428741,\;p_{0}=\mp 0.7157063998,\;q_{0}=\mp 0.2299428741.$

where $\tilde{x}_{0}$ is obtained from
$H(\tilde{x}_{0},\tilde{y}_{0})=\frac{1}{2}\tilde{x_{0}}^{2}-\frac{1}{2}\tilde{y_{0}}^{2}+\frac{1}{4}a_{1}\tilde{y_{0}}^{4}\pm\frac{1}{10}\tilde{y}_{0}=H(0,p_{0}+q_{0}).$
Here, equations (3.16) and (3.17) hold and we evaluate them at $\varphi=\pi$
and $\varphi=\pi,\pi/2$, respectively. We obtain $\gamma_{23}=\pm
0.2464356892\gamma_{33}$ and $\gamma_{31}:=\pm 1.129378222b_{0}.$ A
substitution into the rescaling transformations completes the proof. ∎

######  Remark 3.11 (Equilibria $E_{\pm}$ are initially encircled by limit
cycles $\mathscr{C}^{1}_{\pm}$ and $\mathscr{C}^{2}_{\pm},$ respectively).

When the limit cycles $\mathscr{C}^{1}_{\pm}$ are bifurcated,
$\mathscr{C}^{1}_{+}$ and $\mathscr{C}^{2}_{+}$ encircle the secondary
equilibrium $E_{+}$ while $\mathscr{C}^{1}_{-}$ and $\mathscr{C}^{2}_{+}$
surround $E_{-}$. However, steady-state bifurcations may lead to
disappearances and appearances of $E_{\pm}$ inside these limit cycles. For
instance, Figure 4(h) illustrates unstable limit cycle $\mathscr{C}^{1}_{+}$
encircling spiral sink $E_{-}.$ This is because $\mathscr{C}^{1}_{+}$ and
$\mathscr{C}^{2}_{+}$ are initially bifurcated through a saddle-node
bifurcation of limit cycles and surrounded $E_{+}$ for controller coefficients
from region $b$ in 3(a). The limit cycle $\mathscr{C}^{2}_{+}$ disappears
through a subcritical Hopf bifurcation in Figure 4(c) and next, two equilibria
($E_{-}$ and a saddle) are born through a saddle-node bifurcation at
$T^{-}_{SN}.$ Then, $E_{+}$ and the saddle point disappear at controller
manifold $T^{+}_{SN}.$ This leaves $E_{-}$ as the only equilibrium living
inside the limit cycle $\mathscr{C}^{1}_{+}.$

Now we briefly discuss the differences between symmetry breaking bifurcations
and symmetry preserving bifurcations in the following remark.

######  Remark 3.12 (Symmetry preserving bifurcations versus symmetry breaking
bifurcations).

There are some essential differences between $\mathbb{Z}_{2}$-equivariant
bifurcations of Bogdanov-Takens singularity and its symmetry-breaking
bifurcations. At the start of the analysis, we observe the asymmetric
bifurcations through the appearances of secondary equilibria. Next, each of
these may undergo asymmetric Hopf singularity where two tertiary limit cycles
$\mathscr{C}^{1}_{\pm}$ come to existence; $\mathscr{C}^{1}_{+}$ surrounds
$E_{+}$ while $\mathscr{C}^{1}_{+}$ encircles $E_{-}$. Bifurcated limit cycles
$\mathscr{C}^{1}_{\pm}$ simultaneously collide with the origin and construct a
double homoclinic cycle for the equivariant cases. The symmetry breaking,
instead, causes these to construct two different quaternary homoclinic cycles
$\Gamma_{\pm}$. We have, of course, been naturally looked for these
bifurcations. However, the symmetry-breaking parameters unexpectedly allow the
system to experience two different Bautin bifurcations from $E_{\pm}.$ These
are the alternatives to Hopf bifurcations from $E_{\pm}$ for the symmetric
bifurcations. For Bautin bifurcations, there are an ordered subcritical and
supercritical bifurcations of limit cycles leading to two simultaneous limit
cycles $\mathscr{C}^{1}_{+}$ and $\mathscr{C}^{2}_{+}$. Limit cycle
$\mathscr{C}^{2}_{+}$ has smaller amplitude than $\mathscr{C}^{1}_{+}$ so that
the smaller limit cycle $\mathscr{C}^{2}_{+}$ lives in the interior of
$\mathscr{C}^{1}_{+}$. These limit cycles first merge to a single bistable
limit cycle and then, disappear when controller coefficients cross the saddle-
node controller manifolds of limit cycles. This is different from the saddle-
node bifurcation of limit cycles in [18, Lemma 5.7]. A limit cycle
$\mathscr{C}_{0}$ bifurcates from the primary equilibrium. For the equivariant
bifurcation cases, its symmetric feature causes only a heteroclinic cycle
$\Lambda,$ i.e., it simultaneously collides with the secondary equilibria
$E_{+}$ and $E_{-}.$ For symmetry-breaking bifurcations, the bifurcated limit
cycle $\mathscr{C}_{0}$ collides with either $E_{+}$ or $E_{-}.$ Thus, we have
two different homoclinic cycles $\Lambda_{\pm}$. The heteroclinic cycle
$\Lambda$ is still possible for symmetry-breaking cases by appropriately
tuning the controller coefficients; see Theorem 3.9.

##  4 Nonlinear controllers for linearly uncontrollable cases

Figure 5: Estimated bifurcation controller sets of system (4.1) for the
uncontrollable linearization case

In this section we consider bifurcation control problem for a different
differential system type (1.1) with a single input controller whose
linearization at the origin is either controllable or it is linearly
uncontrollable. Since local symmetry-breaking bifurcations of
$\mathbb{Z}_{2}$-equivariant control systems of type (1.2) are determined by
their cubic Taylor expansion, we consider cubic truncated controlled
differential system (1.2), i.e.,

$\dot{x}:=c_{1}x^{3}+c_{2}y^{3}+c_{3}xy^{2}+c_{4}yx^{2}+u_{1},\;\;\dot{y}:=-x+c_{5}x^{3}+c_{6}y^{3}+c_{7}xy^{2}+c_{8}yx^{2}+u_{2}.$
(4.1)

Here, we only treat single state-feedback input controller systems; i.e., we
choose either $u_{1}:=0$ or $u_{2}:=0.$ When $u_{1}\neq 0$, the linearization
of system (4.1) at the origin is linearly controllable; i.e., it satisfies
Kalman controllability rank condition. For the case $u_{2}\neq 0,$ the
linearized system has an uncontrollable mode. Many nonlinear techniques from
nonlinear control theory fails for systems with uncontrollable modes. In this
section, we show that the available controller coefficients can be exploited
to enforce rich lists of bifurcation scenarios for both controllable and
uncontrollable cases of system (4.1); see Figures 6 and 8.

######  Theorem 4.1 (When linearization satisfies Kalman condition).

Let $u_{2}:=0,$ $u_{1}:=\upsilon_{1}x+{{\upsilon_{2}}}y+{{\upsilon_{3}}}xy,$
and $\upsilon_{i}$ stand for controller coefficients. Then, controlled system
(4.1) admits a Hopf controller manifold, a pitchfork and a heteroclinic
controller manifold given by
$T_{H}=\\{(\upsilon_{1},{{\upsilon_{2}}},{{\upsilon_{3}}})|\,\upsilon_{1}=0\\},$
$T_{P}=\\{(\upsilon_{1},{{\upsilon_{2}}},{{\upsilon_{3}}})|\,{{\upsilon_{2}}}=0\\},$
and

$\displaystyle
T_{HtC}=\Big{\\{}(\upsilon_{1},{{\upsilon_{2}}},{{\upsilon_{3}}})\big{|}\,\frac{\upsilon_{1}}{2}-\frac{c_{3}+3c_{6}}{10c_{2}}{{\upsilon_{2}}}+\frac{120c_{7}+120c_{4}-90c_{8}-270c_{1}}{720}\upsilon_{1}{{\upsilon_{2}}}+\frac{(c_{3}+3c-6)\left(9c_{1}+3c_{8}-4c_{4}-4c_{7}\right)}{60c_{2}}{{{\upsilon_{2}}}}^{2}=0\Big{\\}}.$

Estimated homoclinic controller manifolds $T_{HmC}$ and $T_{HmC_{\pm}}$ in the
controller coefficient space $(\upsilon_{1},\upsilon_{2},\upsilon_{3})$ are

$\displaystyle\frac{\upsilon_{1}}{2}+\frac{77{c_{3}}^{2}+213{c_{6}}^{2}+80c_{2}c_{7}+222c_{3}c_{6}}{320c_{2}(c3+3c_{6}){\upsilon_{1}}^{-2}}+\frac{{c_{3}}^{3}-7c_{6}{c_{3}}^{2}-21{c_{6}}^{2}c_{3}+8c_{3}c_{2}c_{4}+8c_{3}c_{2}c_{7}}{30{c_{2}}^{3}{{{\upsilon_{2}}}}^{-2}}+\frac{135{c_{6}}^{3}+360c_{1}{c_{2}}^{2}+120{c_{2}}^{2}c_{8}-120c_{2}c_{4}c_{6}-408c_{2}c_{6}c_{7}}{240{c_{2}}^{2}(c_{3}+3c_{6}){\upsilon_{1}}^{-1}{{{\upsilon_{2}}}}^{-1}}$
$\displaystyle-\frac{2{{\upsilon_{2}}}}{5c_{2}}(c_{3}+3c_{6})+\frac{27{c_{6}}^{3}-72c_{1}{c_{2}}^{2}-24{c_{2}}^{2}c_{8}+24c_{2}c_{6}c_{4}+24c_{2}c_{6}c_{7}}{30{c_{2}}^{3}}{{{\upsilon_{2}}}}^{2}+\frac{87c_{3}{c_{6}}^{2}-59{c_{3}}^{3}-163c_{6}{c_{3}}^{2}-40c_{2}c_{3}c_{4}-136c_{2}c_{3}c_{7}}{240{c_{2}}^{2}(c_{3}+3c_{6})}\upsilon_{1}{{\upsilon_{2}}}=0,$
$\displaystyle\frac{\upsilon_{1}}{2}-\frac{2{{\upsilon_{2}}}}{5c_{2}}\left(c_{3}+3c_{6}\right)\mp\big{(}\frac{3\sqrt{2}\pi}{32}+\frac{\sqrt{2}\pi\left(21c_{6}+13c_{3}\right)}{480c_{2}}\upsilon_{1}\big{)}{{\upsilon_{3}}}\sqrt{-{{\upsilon_{2}}}}\mp\frac{(450c_{3}c_{6}+891{c_{6}}^{2}-189{c_{3}}^{2}-360c_{2}c_{7})\sqrt{2}\pi}{7680{c_{2}}^{2}}{{\upsilon_{3}}}{{\upsilon_{2}}}\sqrt{-{{\upsilon_{2}}}}=0.$

###### Proof.

Using near-identity changes of coordinates and a Maple program, the system
(4.1) for $u_{2}:=0$ and
$u_{1}:=\upsilon_{0}+\upsilon_{1}x+{{\upsilon_{2}}}y+{{\upsilon_{3}}}xy$ can
be transformed into (1.3) where
$\mu_{1}={{\upsilon_{2}}}+{\scalebox{0.7}{$\mathscr{O}$}}(||\upsilon||^{2}),\mu_{2}=\frac{\upsilon_{1}}{2}+{\scalebox{0.7}{$\mathscr{O}$}}(||\upsilon||^{2}),$

$\displaystyle\mu_{0}=\upsilon_{0}+\frac{288{c_{2}}^{2}c_{8}+864c_{1}{c_{2}}^{2}+77{c_{3}}^{2}c_{6}+231c_{3}{c_{6}}^{2}-288c_{2}c_{4}c_{6}-96c_{2}c_{3}c_{4}-288c_{2}c_{6}c_{7}-96c_{2}c_{3}c_{7}-11{c_{3}}^{3}-297{c_{6}}^{3}}{96{c_{2}}^{2}(c_{3}+3c_{6})}\upsilon_{0}{{\upsilon_{2}}}$
$\displaystyle+\frac{10c_{3}c_{6}+48c_{2}c_{7}+23{c_{3}}^{2}-33{c_{6}}^{2}}{32c_{2}(c_{3}+3c_{6})}\upsilon_{0}\upsilon_{1},$
$\displaystyle\mu_{3}=\frac{1442c_{1}{c_{2}}^{2}+212c_{3}{c_{6}}^{2}+72{c_{3}}^{2}c_{6}+482{c_{2}}^{2}c_{8}-482c_{2}c_{6}c_{7}-162c_{2}c_{3}c_{7}-482c_{2}c_{4}c_{6}-162c_{2}c_{3}c_{4}-272{c_{6}}^{3}-{c_{3}}^{3}}{144{c_{2}}^{2}}\upsilon_{0}+\frac{{{\upsilon_{3}}}}{3}+{\scalebox{0.7}{$\mathscr{O}$}}(||\upsilon||^{2}).$
(4.2)

This implies that the controlled differential system (4.1) is fully unfolded.
Here, we let $\nu_{0}$ for simplicity. The controller curve ${T_{H_{\pm}}}$
are derived by substitution in (3.9). The controller sets for (4.1) when
$\upsilon_{0}=0$ are derived from equations (3.1), (3.10), and (3.27) as
claimed. ∎

(a) The equilibrium is a spiral source. (b) Unstable limit cycle
$\mathscr{C}_{0}$ encircles the spiral sink.

(c) Unstable $\mathscr{C}_{0}$, spiral sinks $E_{\pm}$, saddle origin. (d)
Sink $E_{-}$, source $E_{+}$, origin, unstable $\mathscr{C}_{0},$ and stable
$\mathscr{C}^{1}_{+}$.

(e) Source $E_{+}$, spiral sink $E_{-},$ origin, and unstable
$\mathscr{C}_{0}.$ (f) There are primary saddle, source $E_{+}$ and sink
$E_{-}$.

(g) Two spiral sources $E_{\pm}$, primary saddle, stable $\mathscr{C}^{1}_{-}$
(h) There are two spiral sources $E_{\pm}$ and the primary saddle.

(i) Three equilibria and unstable limit cycle $\mathscr{C}^{1}_{-}$.

Figure 6: Controlled phase portraits for linearly controllable case of (4.1).
Figures 6(a)-6(i) are associated with control coefficients chosen from regions
(a)-(i) in Figure 7(a).

In order to illustrate the numerical $\mathbb{Z}_{2}$-breaking controller
bifurcation varieties, we choose $c_{i}:=1,\,i=1\ldots 8,$ the controller
input ${{\upsilon_{3}}}:=\pm 0.3,$ and obtain Figures 7(a)-7(b). These
numerical controller manifolds are highly accurate over the plotted intervals.
Next, input pairs ($\upsilon_{1},{{\upsilon_{2}}}$) for values
$(0.005,0.005),(-0.005,0.005),$ $(-0.025,-0.0014),$ $(-0.025,-0.0027),$
$(-0.025,-0.005),$ $(-0.025,-0.01),$ $(-0.025,-0.016),$ and $(-0.025,-0.02)$
are chosen from each connected region labeled (a)-(h) in Figure 7(a). We
depict the numerical controlled phase portraits in Figures 6(a)-6(h),
respectively.

######  Theorem 4.2 (Linearly uncontrollable cases).

Let $u_{1}:=0$ and $u_{2}:=\nu_{0}+\nu_{1}y+\nu_{2}y^{2}.$ There are two
saddle node controller manifolds $T_{SN{\pm}}$, two Hopf controller sets
$T_{H{\pm}}$ and two homoclinic controller manifolds $T_{HmC{{\pm}}}$. For
simplicity, let $c_{1}=-1,$ $c_{2}=1,$ $c_{3}=1,$ $c_{4}=-1,$ $c_{5}=-1,$
$c_{6}=1,$ $c_{7}=1,$ and $c_{8}=-1.$ Then, these controller manifolds follow

$\displaystyle
T_{SN{\pm}}=\left\\{(\nu_{0},\nu_{1},\nu_{2})|\,4\nu_{0}\nu_{2}=1\pm
2\nu_{1}+{\nu_{1}}^{2}\right\\},\quad\quad
T_{H{\pm}}=\left\\{(\nu_{0},\nu_{1},\nu_{2})|\,\nu_{1}=\pm
1\mp\sqrt{1+4\nu_{0}\nu_{2}}\right\\},$ $\displaystyle\hbox{ and }\qquad
T_{HmC{{\pm}}}=\left\\{(\nu_{0},\nu_{1},\nu_{2})|\,\frac{1}{2}\nu_{1}+{\frac{16}{45}}{\nu_{0}}^{2}+{\frac{37}{80}}{\nu_{1}}^{2}-\frac{1}{3}\nu_{0}\nu_{2}\mp{\frac{\sqrt{2}\pi\left(16{\nu_{0}}^{2}+3{\nu_{1}}^{2}\right)\left(4\nu_{0}-3\nu_{2}\right)}{32\sqrt{48{\nu_{0}}^{2}+9{\nu_{1}}^{2}}}}=0\right\\}.$

###### Proof.

Here we deal with bifurcation control of system (4.1) when Equations (4.2) is
now replaced with $\mu_{0}=0,$
$\mu_{2}=\frac{3c_{2}c_{4}-c_{3}^{2}}{3c_{2}}\nu_{0}^{2},$
$\mu_{3}=\frac{2(3c_{2}c_{4}+3c_{7}c_{2}-c_{3}^{2}-3c_{3}c_{6})}{9c_{2}}\nu_{0}+\frac{2}{3}\nu_{5},$
and $\mu_{1}$ estimated by

$\displaystyle\mu_{1}=\frac{32\left(3c_{6}+c_{3}\right)\left({c_{3}}^{3}+3c_{6}{c_{3}}^{2}-6c_{2}c_{3}c_{4}-6c_{2}c_{3}c_{7}+27c_{1}{c_{2}}^{2}+9{c_{2}}^{2}c_{8}\right){\nu_{0}}^{2}+9c_{2}\left(64c_{2}c_{4}+80c_{2}c_{7}-23{c_{3}}^{2}-58c_{3}c_{6}+81{c_{6}}^{2}\right){\nu_{1}}^{2}}{576{c_{2}}^{2}(3c_{6}+c_{3})}+\frac{288{c_{2}}\nu_{1}-192c_{3}\nu_{0}\nu_{2}}{576{c_{2}}}.$

The claims are then followed from Theorems 3.2, 3.3, and 3.5. Replacing the
values for $c_{i}$-s, system (4.1) has four equilibria given by

$\displaystyle(x_{1}^{\pm},y_{1}^{\pm})=\left(\frac{1+\nu_{1}\pm\sqrt{1+2\nu_{1}+\nu_{1}^{2}-4\nu_{0}\nu_{2}}}{2\nu_{2}},\frac{1+\nu_{1}\pm\sqrt{1+2\nu_{1}+\nu_{1}^{2}-4\nu_{0}\nu_{2}}}{2\nu_{2}}\right),$
$\displaystyle(x_{2}^{\pm},y_{2}^{\pm})=\left(\frac{1-\nu_{1}\pm\sqrt{1-2\nu_{1}+\nu_{1}^{2}-4\nu_{0}\nu_{2}}}{2\nu_{2}},\frac{1-\nu_{1}\pm\sqrt{1-2\nu_{1}+\nu_{1}^{2}-4\nu_{0}\nu_{2}}}{2\nu_{2}}\right).$

The estimated controller sets follow equations (3.5), (3.9), and (3.10). ∎

We take $\nu_{1}=0.1$ and depict critical controller manifolds in terms of
controller coefficients ($\nu_{0},\nu_{2}$) in Figures 5. We choose
$(-0.7,-0.7),$ $(-0.4,-0.6),$ $(0.1,-0.2),$ $(0.1,-0.5),$ $(0.1,-0.66),$ and
$(0.3,-0.1)$ for $(\nu_{0},\nu_{2})$ from regions 1\- 6 in the first figure of
5. Then, the numerical phase portraits are illustrated in Figures 8(a)-8(e),
respectively. System (4.1) has no equilibrium for controller coefficients
choices from region 1. Two small equilibria are bifurcated via a fold
bifurcation at $T_{SN{+}}$; see Figure 8(b). Another saddle-node bifurcation
occurs at $T_{SN{-}}$ and two new equilibria are born. Thus, there are four
equilibria for controller coefficient choices from region 3. As for controller
coefficients of region 4, a subcritical Hopf bifurcation gives rise to an
unstable limit cycle; see Figure 8(d). The limit cycle disappears at
homoclinic controller set $T_{HmC{{+}}}$ and therefore, there is no limit
cycle for coefficients from region 5. Control choices associated with region 6
lead to a stable limit cycle.

######  Remark 4.3 (Subcritical and supercritical type switching for
bifurcation varieties).

Linear stability analysis provides a local approach and its neighborhood
validity is essential for practical life problems. The neighborhood validity
can be very small if nonlinear terms give rise to a subcritical Hopf
bifurcation. The basin of attraction for the asymptotically stable equilibrium
in subcritical cases is merely the interior of the small bifurcated unstable
limit cycle. Therefore, the linear stability of this kind fails in many
control engineering applications. Primary supercritical Hopf bifurcation has
been widely considered as a safe controller design; e.g., see [26]. This is
due to its expected large basin of attraction for the bifurcated
asymptotically stable limit cycle.

Figure 5 illustrates that controller coefficient choices from region 4
correspond with unstable limit cycle in Figure 8(d) while choices from region
6 cause stable limit cycle in Figure 8(f). The same phenomenon occurs for
regions $g$ and $i$ in Figure 7(a). We have stable limit cycle
$\mathscr{C}^{1}_{-}$ in Figure 6(g) while $\mathscr{C}^{1}_{-}$ is unstable
in Figure 6(i). They play a potential role for stabilizing the system.
Controller coefficients from regions 6 and $i$ from 6(i) and 7(a) fail the
restrictions in Theorem 3.3. These imply the existence of Bautin bifurcation
in Theorem 3.7.

(a) ${{\upsilon_{3}}}:=0.3$, $a_{1}=b_{0}=1$ (b) ${{\upsilon_{3}}}:=-0.3,$
$a_{1}=b_{0}=1$

(c) ${\upsilon_{3}}:=\pm 0.3,a_{1}=b_{0}=-1$ (d) ${{\upsilon_{3}}}:=\pm 0.3,$
$b_{0}=-a_{1}=1$

Figure 7: Symmetry-breaking controller set for linearly controllable case of
(4.1), i.e., $u_{2}:=0.$

(a) No equilibrium (b) Two equilibrium

(c) No stable equilibrium (d) An unstable limit cycle

(e) One stable equilibrium (f) A stable limit cycle

Figure 8: Controlled phase portraits 8(a)-8(e) corresponding with regions 1-5
partitioned by the estimated controller sets in Figure 5 for uncontrollable
linearization case of (4.1)

##  5 Chua system

The main goal in this section is to illustrate how controlled Chua system
experiences a rich list of bifurcation scenarios by appropriately choosing
small controller coefficients. There have been an extensive literature on the
dynamics study of Chua circuit systems; e.g., see [34, 40, 43]. For example,
Zhao et. al. [43] investigated Chua systems for Hopf and generalized Hopf
bifurcations. Yang and Zhao [40] considered the modifed Chua’s circuit system
with a delayed feedback: a delayed system undergoes Hopf and Hopf-zero
bifurcations and stabilized the system through either a stable periodic orbit
or a stable equilibrium. Puebla et. al. [34] implemented a linear PI
compensator for a tracking control. Chua circuit is an electrical circuit and
experiences Bogdanov-Takens, Hopf-zero and Hopf bifurcations. Consider the
controlled Chua system

$\displaystyle\dot{x}=\alpha(y-ax^{3}-cx),\quad\dot{y}=x-y+z,\quad\dot{z}=-\beta
y+u,\quad u=\nu_{0}+\nu_{1}x+\nu_{2}y+\nu_{3}xy,$ (5.1)

where $u$ stands for a linear state-feedback input with small gain parameters
$\nu_{i}$ for $i=0,1,2,3$. State variables $x$ and $y$ represent the voltage
of the capacitor and $z$ is the electric current in the inductor. The
uncontrolled system $(\nu_{i}=0\hbox{ for }i\leq 4)$ is
$\mathbb{Z}_{2}$-equivariant whose symmetry is given by the reflection
$(x,y,z)\longrightarrow(-x,-y,-z)$. Uncontrolled Chua circuit system has three
meaningful equilibria: the origin and
$e_{\pm}:\left(x_{\pm},y,z_{\pm}\right)=\left(\pm\sqrt{-\frac{c}{a}},0,\mp\sqrt{-\frac{c}{a}}\right).$
The origin experiences Hopf and Hopf-zero bifurcations when
$(\beta=-\alpha(c-1)(1+\alpha c),0<c<1)$ and
$(\beta=0,\alpha=-\frac{1}{c},0<c<1)$, respectively. The equilibria $e_{\pm}$
undergo Hopf and Hopf-zero bifurcations for $(\beta=\alpha(2c+1)(1-2\alpha
c),-\frac{1}{2}<c<0)$ and $(\beta=0,\alpha=\frac{1}{2c},-\frac{1}{2}<c<0)$.
These four singularities are not treated here; see [40, 34]. We remark that
our Hopf bifurcations here are caused by controller coefficient choices and
thus, they are essentially different from Hopf singular cases caused by the
system’s choices for $\alpha,\beta$ and $c$.

######  Proposition 5.1.

When $c=0,$ the origin is the only equilibrium of the system. System (5.1) has
a $\mathbb{Z}_{2}$-equivariant Bogdanov-Takens singularity at the origin for
$\beta=\alpha$ and $c=0$.

###### Proof.

Jacobian matrix associated with (5.1) at $(0,0)$ for $u=0$ is $J:=[-\alpha
c,\alpha,0;1,-1,1;0,-\beta,0].$ Eigenvalues for $c=0$ are $0$ and
$-\frac{1\pm\sqrt{1+4\alpha-4\beta}}{2}$. Thus for $\alpha=\beta,$ we have a
none semi-simple double zero eigenvalue. ∎

######  Theorem 5.2 (Controlled bifurcations).

Consider controlled Chua differential system (5.1) for $\beta=\alpha$ and
$c=0$. Then by varying small controller coefficients $\nu_{i}$, the controlled
system undergoes a pitchfork bifurcation, three Hopf bifurcations and two
homoclinic bifurcations.

###### Proof.

By Proposition 5.1, there is a two-dimensional invariant center manifold
$\mathscr{M}$ for Chua system (5.1). The reduction of (5.1) on center manifold
$\mathscr{M}$ and then, primary shift of coordinates give rise to
$\dot{x}=v_{1}+a\alpha(\alpha-1)^{3}x^{3}+3a\alpha^{2}(\alpha-1)^{2}x^{2}y+a\alpha^{4}y^{3}$
and $\dot{y}=-x+\nu_{3}xy+v_{2},$ for

The first level normal form is obtained through a finite sequence of flow
time-one maps generated by the initial value problems
$\dot{x}\frac{\partial}{\partial x}+\dot{y}\frac{\partial}{\partial y}=T_{i},$
$x(0,\nu,X,Y)=X,$ and $y(0,\nu,X,Y)=Y$ for $i=1,2,3.$ Further, we assign each
homogenous monomial vector field with a grade equal to its degree minus one
plus two times degree of its parameters, e.g.,
$d(x^{2}y^{3}\nu_{1}{\nu_{2}}^{3})=2+3+2+6=13.$ The image of homological
operator $L^{k}:\mathscr{L}_{k}\rightarrow\mathscr{L}_{k}$ defined by
$L^{k}(w):=[w,-x\frac{\partial}{\partial y}]$ can be used to simplify the
system. Hence, we consider the Lie brackets given by

$\displaystyle\left[y\frac{\partial}{\partial x},-x\frac{\partial}{\partial
y}\right]=x\frac{\partial}{\partial x}-y\frac{\partial}{\partial
y},\left[\frac{1}{2}x\frac{\partial}{\partial
x}-\frac{1}{2}y\frac{\partial}{\partial y},-x\frac{\partial}{\partial
y}\right]=-x\frac{\partial}{\partial y},\left[y^{3}\frac{\partial}{\partial
x},-x\frac{\partial}{\partial y}\right]=3xy^{2}\frac{\partial}{\partial
x}-y^{3}\frac{\partial}{\partial y},$
$\displaystyle\left[\frac{1}{2}x^{2}y\frac{\partial}{\partial
x}-\frac{1}{2}xy^{2}\frac{\partial}{\partial y},-x\frac{\partial}{\partial
y}\right]=\frac{1}{2}x^{3}\frac{\partial}{\partial
x}-\frac{3}{2}x^{2}y\frac{\partial}{\partial
y},\left[x^{2}y\frac{\partial}{\partial x}+xy^{2}\frac{\partial}{\partial
y},-x\frac{\partial}{\partial y}\right]=x^{3}\frac{\partial}{\partial
x}+x^{2}y\frac{\partial}{\partial y},$ (5.2)
$\displaystyle\left[\frac{3}{4}xy^{2}\frac{\partial}{\partial
x}-\frac{1}{4}y^{3}\frac{\partial}{\partial y},-x\frac{\partial}{\partial
y}\right]=\frac{3}{2}x^{2}y\frac{\partial}{\partial
x}-\frac{3}{2}xy^{2}\frac{\partial}{\partial
y},\left[xy^{2}\frac{\partial}{\partial x}+y^{3}\frac{\partial}{\partial
y},-x\frac{\partial}{\partial y}\right]=2x^{2}y\frac{\partial}{\partial
x}+2xy^{2}\frac{\partial}{\partial y}.$

Hence, terms of the form $x^{3}\frac{\partial}{\partial
x},x^{2}y\frac{\partial}{\partial y},$ $x^{2}y\frac{\partial}{\partial
x},xy^{2}\frac{\partial}{\partial y}$ and parametric terms associated with
$x\frac{\partial}{\partial y}$ can be simplified from the system. However, we
can choose between $xy^{2}\frac{\partial}{\partial x}$ or
$y^{3}\frac{\partial}{\partial y}$ to simplify from the system. This is also
true for parametric terms $x\frac{\partial}{\partial x}$ and
$y\frac{\partial}{\partial y},$ where only one of them can be simplified from
the system. Given the Lie brackets in equation (5.2) and the grading function,
we choose $T_{1}$ to simplify terms of grade 2 as follows

$\displaystyle
T_{1}^{x}=\left(\frac{1}{2}\alpha\nu_{1}-\nu_{1}-\frac{1}{2}\nu_{2}\right)y-{\frac{\left(\alpha\nu_{1}-\nu_{1}-\nu_{2}\right)x}{2\alpha}}-a\alpha\,\left(\alpha-1\right)^{3}{x}^{2}y-\frac{3a{\alpha}^{2}\left(\alpha-1\right)^{2}xy^{2}}{2}-\frac{3a{\alpha}^{3}\left(\alpha-1\right)y^{3}}{4},$
$\displaystyle
T_{1}^{y}={\frac{\left(\alpha\nu_{1}-\nu_{1}-\nu_{2}\right)y}{2\alpha}}-\frac{a\alpha\,\left(\alpha-1\right)^{3}xy^{2}}{2}-\frac{a{\alpha}^{2}\left(\alpha-1\right)^{2}y^{3}}{2}.$

The updated system is given by $\dot{X}\frac{\partial}{\partial
X}+\dot{Y}\frac{\partial}{\partial Y}=\exp{\rm
ad}_{T_{1}^{x}\frac{\partial}{\partial x}+T_{1}^{y}\frac{\partial}{\partial
y}}(v_{1}\frac{\partial}{\partial x}+v_{2}\frac{\partial}{\partial y})$ whose
grade two terms are associated with

$\displaystyle\left(a\alpha^{4}Y^{3}-\alpha\nu_{1}Y-\frac{\alpha\nu_{1}-\nu_{2}}{2}X+\frac{3a\alpha^{3}(\alpha-1)}{4}XY^{2}\right)\frac{\partial}{\partial
X}+\left(\frac{3a\alpha^{3}(\alpha-1)}{4}Y^{3}-\frac{\left(\alpha\nu_{1}-\nu_{2}\right)}{2}Y\right)\frac{\partial}{\partial
Y}$

For simplicity of notations, we replace $(X,Y)$ with $(x,y).$ Due to the Lie
brackets

$\displaystyle\left[\frac{1}{3}x^{2}\frac{\partial}{\partial
x}-\frac{2}{3}xy\frac{\partial}{\partial y},-x\frac{\partial}{\partial
y}\right]=-x^{2}\frac{\partial}{\partial
y},\left[\frac{2}{3}xy\frac{\partial}{\partial
x}-\frac{1}{3}y^{2}\frac{\partial}{\partial y},-x\frac{\partial}{\partial
y}\right]=\frac{2}{3}x^{2}\frac{\partial}{\partial
x}-\frac{4}{3}xy\frac{\partial}{\partial y}$
$\displaystyle\left[xy\frac{\partial}{\partial
x}+y^{2}\frac{\partial}{\partial y},-x\frac{\partial}{\partial
y}\right]=x^{2}\frac{\partial}{\partial x}+xy\frac{\partial}{\partial
y},\left[y^{2}\frac{\partial}{\partial x},-x\frac{\partial}{\partial
y}\right]=2xy\frac{\partial}{\partial x}-y^{2}\frac{\partial}{\partial y},$

we can eliminate all of the terms $x^{2}\frac{\partial}{\partial y},$
$xy\frac{\partial}{\partial y},$ $x^{2}\frac{\partial}{\partial x},$ while
only one of $xy\frac{\partial}{\partial x}$ and $y^{2}\frac{\partial}{\partial
y}$ can be simplified. The calculations show that only the following terms of
grade 3 remain in the system

$\displaystyle\Big{(}\frac{3\alpha^{3}a\nu_{0}\left(1+\alpha^{2}\right)y^{2}-\nu_{0}\left(\nu_{2}-2\alpha\nu_{2}+\nu_{1}-\alpha\nu_{1}+2\alpha^{2}\nu_{1}\right)}{2\alpha}+\frac{\alpha\left(3\nu_{0}a\left(\alpha-1\right)\left(1+{\alpha}^{2}\right)+\nu_{3}\right)xy}{3}\Big{)}\frac{\partial}{\partial
x}+\frac{\alpha\left(3a\nu_{0}\left(\alpha-1\right)\left({\alpha}^{2}+1\right)+\nu_{3}\right)y^{2}}{3}\frac{\partial}{\partial
y}$

Let $A^{-1}_{r-i}(y):=y^{r-i+1}\frac{\partial}{\partial x}.$ Then,

$\displaystyle A^{-1}_{r}(y+f(\nu))=\sum_{i=0}^{r+1}{r+1\choose
i}f(\nu)^{i}y^{r-i+1}\frac{\partial}{\partial x}.$

Thus, term
$\frac{3}{2}\alpha^{2}a(1+\alpha^{2})\nu_{0}y^{2}\frac{\partial}{\partial x}$
can be cancelled by replacing $y$ with $y+f(\nu)$ where
$f(\nu)=-\frac{1}{2}\alpha^{2}a(1+\alpha^{2})\nu_{0}$. Thereby, the system is
normalized to (1.3), where
$\mu_{0}=\frac{9}{8}\alpha\nu_{1}\nu_{0}-\frac{3}{32}\nu_{1}\nu_{0}-\frac{33}{32\alpha}\nu_{1}\nu_{0}-\frac{37}{32}\nu_{2}\nu_{0}-\frac{11}{32\alpha}\nu_{2}\nu_{0}-\nu_{0},$

$\displaystyle\mu_{1}=-\frac{5\alpha^{2}\nu_{1}^{2}}{16}-\frac{3\nu_{1}^{2}}{4}-\frac{3\alpha\nu_{1}^{2}}{16}-\frac{3\alpha\nu_{1}\nu_{2}}{16}-\frac{5\nu_{1}\nu_{2}}{16}-\frac{\nu_{2}^{2}}{4}-\frac{1047}{5120}a\alpha^{4}\nu_{0}^{2}+\frac{2583}{1280}a\alpha^{3}\nu_{0}^{2}-\frac{1047}{5120}a\nu_{0}^{2}-\frac{1857}{512}a\alpha^{2}\nu_{0}^{2}+\frac{2583}{1280}a\alpha\nu_{0}^{2}-\alpha\nu_{1},\qquad$
$\displaystyle\mu_{2}=-\frac{49\alpha^{2}+3\alpha+12}{64}\nu_{1}^{2}+\frac{27\alpha^{2}-\alpha+6}{32\alpha}\nu_{1}\nu_{2}-\frac{5(\alpha-1){\nu_{2}}^{2}}{64\alpha}+\frac{(63\alpha-31)(\alpha-1)\nu_{3}\nu_{0}}{64\alpha}+\frac{3a(2829\alpha^{2}-3226\alpha+2829)(\alpha-1)^{3}{\nu_{0}}^{2}}{10240\alpha}-\frac{\alpha\nu_{1}}{2}+\frac{\nu_{2}}{2},$
$\displaystyle\mu_{3}=\frac{1}{3}\alpha\nu_{3}-\frac{9}{16}\alpha^{3}a\nu_{0}+\frac{9}{16}a\alpha^{2}\nu_{0}-\frac{3}{16}a\alpha\nu_{0}+\frac{3}{16}a\alpha^{4}\nu_{0},\qquad\qquad\hbox{
and }\qquad\qquad a_{1}=a\alpha^{4},\quad
b_{0}=\frac{3}{4}a(\alpha-1)\alpha^{3}.$

For $\nu_{0}=0,$ $\mu_{0}=0.$ Thus, we follow Proposition 3.1, Theorem 3.3,
and Theorem 3.5 for deriving symbolic estimates of $T_{P},$ $T_{H},$
$T_{H\pm}$ and $T_{HmC{{\pm}}}.$ For simplicity of the formulaes, we take
$\nu_{3}=0.3,$ $\alpha:=0.8,$ $a:=1.$ Then, we obtain the following equations:

$\displaystyle
T_{H}=\left\\{(\nu_{1},\nu_{2})\big{|}\,\nu_{1}=\frac{5}{4}\nu_{2}\right\\},\quad
T_{HmC}=\left\\{(\nu_{1},\nu_{2})\big{|}\,\nu_{1}=\frac{320}{1771}+\frac{35045}{28336}\nu_{2}-\frac{5}{28336}\sqrt{1048576+5286912\nu_{2}+13861929{\nu_{2}}^{2}}\right\\},$
$\displaystyle
T_{p}=\left\\{(\nu_{1},\nu_{2})\big{|}\,\nu_{1}=0\right\\},\quad
T_{HmC_{\pm}}=\left\\{(\nu_{1},\nu_{2})\big{|}\,\pm\frac{4989\sqrt{10}}{1600000}\pi\nu_{1}\sqrt{\nu_{1}}\pm\frac{21\sqrt{10}}{160000}\pi\sqrt{\nu_{1}}\nu_{2}-\frac{16}{25}\nu_{1}+\frac{1}{2}\nu_{2}\pm\frac{9\sqrt{10}}{1000}\pi\sqrt{\nu_{1}}=0\right\\},$
$\displaystyle
T_{H_{\pm}}=\Big{\\{}(\nu_{1},\nu_{2})\big{|}\,\frac{5293162496}{30517578125}{\nu_{2}}^{2}-\frac{67273949184}{152587890625}\nu_{1}\nu_{2}\pm\frac{978767872\sqrt{5}}{30517578125}\sqrt{\nu_{1}}\nu_{2}-\frac{1841299456}{30517578125}\nu_{1}$
$\displaystyle+\frac{2097152}{48828125}\nu_{2}\pm\frac{199753728\sqrt{5}}{30517578125}\sqrt{\nu_{1}}=0\Big{\\}}.$

These are derived from equations (3.1), (3.9), and (3.10), respectively. ∎

(a) $\mathbb{Z}_{2}$-breaking controller sets for controlled Chua system (5.1)
when $\nu_{3}=0.3,$ $\nu_{0}=0,$ $\alpha:=0.8,$ and $a:=1$

(b) Trajectories and regulating controller $u$ associated with region (a) for
$\nu_{1}=-0.01,$ $\nu_{2}=-0.02.$

(c) Trajectories and controller $u$ for region (b) when
$(\nu_{1},\nu_{2})=(-0.01,0)$.  (d) Region (b) for
$(\nu_{1},\nu_{2})=(-0.02,0).$

Figure 9: Regularization of the origin using bifurcation control and
stabilization via supercritical Hopf bifurcation; controller coefficients are
from regions (a) and (b) in Figure 9(a) when $\alpha:=0.8,$ $a:=1,$
$\nu_{3}=0.3,$ and $\nu_{0}=0$.

Let $\nu_{3}=0.3$ and $\nu_{0}=0$. Then, controller coefficient choices
$(\nu_{1},\nu_{2})$ from regions (a)-(h) from Figure 9(a) give rise to the
controlled trajectories of Chua system (5.1) in Figures 9(b)-9(d),
10(a)-10(d), and 11(a)-11(b), respectively.

###  5.1 Controller objectives

Different bifurcation scenarios can be realised by appropriate choices of
controller coefficients and these provide an effective approach to meet many
desired control objectives. For example, we explain how these objectives can
include feedback regularization, stabilization, amplitude size and frequency
control for oscillations of the controlled differential Chua system.

#### Region (a): A feedback regularization by bifurcation control:

Controller coefficient choices from region (a) in Figure 9(a) turn the origin
into a spiral sink. Hence, the feedback control $u$ in (5.1) regularizes the
origin and all trajectories converge to the origin; the origin is globally
asymptotically stable. Figure 9(b) depicts trajectories $x(t),$ $y(t),$ and
$z(t)$ for initial values $(-0.6,-0.54,-0.1)$ and controller coefficients
$(\nu_{1},\nu_{2})=(-0.01,-0.02)$ chosen from region (a). These trajectories
converge to the origin while the controller $u$ is very small.

#### Region (b): Stabilization and oscillation control via a supercritical
Hopf bifurcation:

There is a stable limit cycle $\mathscr{C}_{0}$ for region $b$ and
trajectories converge to $\mathscr{C}_{0}.$ By Proposition 3.1, there is a
supercritical Hopf bifurcation at the border between regions $a$ and $b,$ that
is estimated by $\nu_{1}=\frac{5}{4}\nu_{2}.$ This provides a stabilization
approach for the system. Furthermore, the leading estimate for radius of the
bifurcation limit cycle is given by
$\sqrt{\dfrac{4(\alpha\nu_{1}-\nu_{2})}{3a(\alpha-1)\alpha^{3}}}$ while the
leading estimate for its angular frequency is $\sqrt{\alpha\nu_{1}}.$ Figures
9(c)-9(d) illustrate converging trajectories to the limit cycle
$\mathscr{C}_{0}$ from the initial values $(0.3,0.54,0.1)$ for controller
coefficient choices $(\nu_{1},\nu_{2})$ from region (b). Since the angular
frequency and amplitude size of oscillations are proportional to the angular
frequency and radius of the bifurcation limit cycle, we can control the
oscillating trajectories accordingly. Hence, an increase in $\nu_{1}$ leads to
an increase in angular frequency of oscillations while
$|\alpha\nu_{1}-\nu_{2}|$ is the deciding factor for the amplitude sizes of
the oscillations. Therefore, an increase from $|\nu_{1}|=0.01$ to
$|\nu_{1}|=0.02$ gives rise to an amplification in both amplitude sizes and
angular frequencies in figures 9(c)-9(d). Notice that the controller $u$ in
figure 9(c) is very small while it causes oscillations with moderate
magnitudes.

(a) Region (c) for $(\nu_{1},\nu_{2})=(0.01,0.1)$ (b) Region (d) for
$(\nu_{1},\nu_{2})=(0.02,0.068)$

(c) Region (e) for $(\nu_{1},\nu_{2})=(0.02,0.03)$ (d) Region (f) for
$(\nu_{1},\nu_{2})=(0.02,0)$

Figure 10: Controlled trajectories 10(a)-10(d) associated with regions (c)-(f)
from Figure 9(a) for (5.1), $\alpha:=0.8,$ $a:=1$, $\nu_{3}=0.3,$ $\nu_{0}=0$.

#### Region (c): A pitchfork bifurcation of equilibria inside
$\mathscr{C}_{0}$ within center manifold $\mathscr{M}$.

There is a pitchfork bifurcation at border of regions (b) and (c). The system
for controller coefficients from region (b) has a limit cycle
$\mathscr{C}_{0}.$ The limit cycle $\mathscr{C}_{0}$ lives on the center
manifold $\mathscr{M}$ of Chua system and within the center manifold
$\mathscr{M}$, $\mathscr{C}_{0}$ encircles the origin. As the controller
coefficients cross the pitchfork variety $T_{p},$ two new equilibria $e_{\pm}$
are born from the origin. Hence, for controller coefficients chosen from
region (c), controlled Chua system (5.1) has two spiral sources, and origin as
a hyperbolic saddle. All these equilibria live inside the stable limit cycle
$\mathscr{C}_{0}$ within the center manifold. Trajectories in Figure 10(a)
converge to the asymptotically stable limit cycle $\mathscr{C}_{0}$ for
initial values $(-0.7,-0.7,-0.3)$ and $(\nu_{1},\nu_{2})=(0.01,0.1).$ The
oscillation frequency and amplitude size control of the oscillating
trajectories are controlled by the management of the stable limit cycle
$\mathscr{C}_{0}$. Limit cycle $\mathscr{C}_{0}$ is controllable as similar to
the controller coefficient cases from region (b).

#### Region (d): A secondary subcritical Hopf bifurcation of
$\mathscr{C}^{1}_{-}$ inside $\mathscr{C}_{0}$ within $\mathscr{M}$.

By Theorem 3.3, a secondary Hopf bifurcation of limit cycles from $e_{-}$
occurs inside the limit cycle $\mathscr{C}_{0}$. Remark that equilibria
$e_{\pm},$ the origin, and limit cycles $\mathscr{C}^{1}_{-}$ and
$\mathscr{C}_{0}$ all live on the center manifold $\mathscr{M}$. Hence for
controller coefficient choices from region (d), the controlled system (5.1)
has a spiral source, a saddle, a spiral sink and two limit cycles. The leading
estimates for angular frequency and radius of $\mathscr{C}^{1}_{-}$ are given
by $\frac{\sqrt{2}}{4}(16\alpha\nu_{1}-(\nu_{2}-\alpha\nu_{1})^{2})$ and
$-\frac{8\sqrt{2}}{7a\alpha^{4}}\sqrt{\frac{14}{3(\alpha-1)}(\nu_{2}-\alpha\nu_{1})-184((\nu_{2}-\alpha\nu_{1})^{2}-4\alpha\nu_{1})}-\frac{32}{7}\frac{\sqrt{4\alpha\nu_{1}-(\nu_{2}-\alpha\nu_{1})^{2}}}{a_{1}},$
respectively. However, $\mathscr{C}^{1}_{-}$ is an unstable limit cycle.
Hence, amplitude size control of the oscillating dynamics is typically
determined by the stable limit cycle $\mathscr{C}_{0}$. Figure 10(b)
represents controlled dynamics associated with region (d). Controlled
trajectories for initial values $(0.004,-0.1,0)$ converge to $e_{-}$ when
controller coefficients $(\nu_{1},\nu_{2})=(0.02,0.068)$ are taken from region
(d).

#### Region (e): Disappearance of $\mathscr{C}^{1}_{-}$ via homoclinic
bifurcation at $T_{HmC-}$.

The limit cycle $\mathscr{C}^{1}_{-}$ collides with the origin and constructs
a homoclinic cycle ${\Gamma}_{-}$ exactly when controller coefficients are
taken from homoclinic controller set $T_{HmC-};$ see Theorem 3.5. Therefore,
both of $\mathscr{C}^{1}_{-}$ and ${\Gamma}_{-}$ disappear when we take
controller coefficients from region (e) in Figure 9(a). Hence, there exist a
source, a saddle, a sink and an enlarged stable limit cycle $\mathscr{C}_{0}.$
Trajectories in Figure 10(c) for initial values $(-0.015,-0.001,0.015)$
converge to $e_{-}$ for $(\nu_{1},\nu_{2})=(0.02,0.03)$ from region (e).

#### Controlled dynamics of region (f): $\mathscr{C}_{0}$ disappears via
homoclinic bifurcation at $T_{HmC}$.

The limit cycle $\mathscr{C}_{0}$ collides with origin for controller
coefficients taken from transition variety $T_{HmC}$. Hence, limit cycle
$\mathscr{C}_{0}$ disappears from the local dynamics of controlled Chua system
when we take controller coefficients from region (f). Therefore, there are a
source, a saddle, a sink and no limit cycle. Figure 10(d) depicts a trajectory
that converges to $e_{-}$ from initial values $(-0.02,-0.001,0.1)$ for
controllers $(\nu_{1},\nu_{2})=(0.02,0)$ corresponding with region (f). For
controller coefficients from region (f), system (5.1) has three equilibria and
no limit cycle.

(a) Region (g): $\nu_{1}=0.018,$ $\nu_{2}=-0.016.$ Orbits converge to
$\mathscr{C}_{+}$ and $e_{-}.$

(b) Region (h): $\nu_{1}=0.02,$ $\nu_{2}=-0.025.$ Trajectories converge to
$e_{\pm}.$

Figure 11: Controlled trajectories 11(a)-11(b) correspond with regions (g) and
(h) from Figure 9(a) for (5.1), $\alpha:=0.8,$ $a:=1$, $\nu_{3}=0.3,$
$\nu_{0}=0$.

#### Region (g): Limit cycle $\mathscr{C}^{1}_{+}$ appears when controller
coefficients cross $T_{HmC+}$.

By Theorem 3.5, there is a stable limit cycle $\mathscr{C}^{1}_{+}$ for
controller coefficients from region (g), i.e., there is a homoclinic variety
$T_{HmC+}$ at the border between regions (f) and (g). In fact, there is a
homoclinic cycle ${\Gamma}_{+}$ (encircling $e_{+}$) for controller
coefficients from $T_{HmC+}$. There are a stable limit cycle
$\mathscr{C}^{1}_{+}$ encircling unstable equilibrium $e_{+},$ and two more
equilibria outside of $\mathscr{C}^{1}_{+}$. The latter equilibria are a
saddle and a sink $e_{-}$. All equilibria, homoclinic cycle ${\Gamma}_{+}$ and
limit cycle $\mathscr{C}^{1}_{+}$ live on center manifold $\mathscr{M}.$
Trajectories in Figures 11(a) converge to the stable limit cycle
$\mathscr{C}^{1}_{+}$ and $e_{-}$ for controller coefficients $\nu_{1}=0.018$
and $\nu_{2}=-0.016$ from region (g), initial values $(0.02,0.005,-0.02)$ and
$(-0.015,-0.001,0.3),$ respectively.

#### Region (h): Limit cycle $\mathscr{C}^{1}_{+}$ shrinks in size to coalesce
with $e_{+}$ and disappear.

Theorem 3.3 concludes that there is a supercritical Hopf bifurcation at
$T_{H+},$ where $\mathscr{C}^{1}_{+}$ is born from $e_{+}.$ Controller
coefficients crossing $T_{H+}$ and entering region (h) lead to change of
stability for $e_{+}$ and disappearance of $\mathscr{C}^{1}_{+}.$ Hence for
controller coefficients from region (g), there exist two stable equilibria
$e_{\pm},$ a saddle origin and no limit cycle. Trajectories in Figures 11(b)
for initial values $(0.013,-0.03,0.5)$ and $(-0.013,-0.001,0.1)$ converge to
$e_{+}$ and $e_{-},$ respectively.

######  Remark 5.3 (Stabilization and regularization for controller
coefficient choices from regions (g) and (h)).

Controller coefficients associated with region (h) lead to two asymptotically
stable equilibria $e_{+}$ and $e_{-},$ whose their combined basins of
attraction is the whole state space except the stable and unstable manifolds
of the saddle origin. Therefore, controller coefficients chosen from region
(h) give a regularization approach for the system, where the controlled
trajectories converge to equilibria close to the origin; also see [31] for a
similar phenomenon. Since Hopf bifurcation at the border between regions (h)
and (g) is supercritical, controller coefficient choices from region (g) have
a stabilization property. In fact, every trajectory converges to either
$\mathscr{C}^{1}_{+}$ or $e_{-},$ except that a trajectory would exactly fall
on the stable and unstable manifolds of the saddle origin. The latter is
practically impossible due to imperfections, errors, etc., and the fact that
the stable and unstable manifolds are structurally unstable and have a zero-
Lebesgue measure.

##  6 Conflict of interest

There is no conflict of interest to report.

##  7 Data availability statement

There is no data associated with this paper to report.

## References

  * [1] A. Algaba, K.W. Chung, B.W. Qin, A.J. Rodriguez-Luis, Computation of all the coefficients for the global connections in the $\mathbb{Z}_{2}$-symmetric Takens-Bogdanov normal forms, Commun. Nonlinear Sci. Numer. Simulat. 81 (2020) 105012.
  * [2] A. Algaba, K.W. Chung, B.W. Qin, A.J. Rodriguez-Luis, A nonlinear time transformation method to compute all the coefficients for the homoclinic bifurcation in the quadratic Takens–Bogdanov normal form, Nonlinear Dyn. 97 (2019) 979–990.
  * [3] A. Algaba, M. Merino, A.J. Rodriguez-Luis, Homoclinic interactions near a triple-zero degeneracy in Chua’s equation, Int. J. Bifur. Chaos 22 (2012), no. 6, 1250129, 16 pp.
  * [4] A. Algaba, M. Merino, F. Fernandez-Sánchez, A.J. Rodriguez-Luis, Hopf bifurcations and their degeneracies in Chua’s equation, Int. J. Bifur. Chaos 21 (2011), no. 9, 2749–2763.
  * [5] A. Algaba, F. Fernández-Sánchez, M. Merino, A.J. Rodriguez-Luis, Analysis of the T-point-Hopf bifurcation with $\mathbb{Z}_{2}$-symmetry. Application to Chua’s equation, Int. J. Bifur. Chaos 20 (2010) 979–993.
  * [6] M.R. Candido, D.D. Novaes, C. Valls, Periodic solutions and invariant torus in the Rössler system, Nonlinearity 33 (2020) 4512–4538.
  * [7] T.R. Blows, L.M. Perko, Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems, SIAM Review 36 (1994) 341–376.
  * [8] G. Chen, D.J. Hill, X. Yu, Bifurcation Control Theory and Applications, Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin 2003.
  * [9] G. Chen, J.L. Moiola, H.O. Wang, Bifurcation control: theories, methods and applications, Internat. J. Bifur. Chaos 10 (2000) 511–548.
  * [10] G. Gaeta, Bifurcation and symmetry breaking, Phys. Rep. 189 (1990) 1–87.
  * [11] G. Gaeta, Further reduction of Poincare-Dulac normal forms in symmetric systems, Cubo 9 (2007) 1–11.
  * [12] G. Gaeta, Poincare renormalized forms, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999) 461–514.
  * [13] M. Gazor, M. Kazemi, Singularity: A Maple library for local bifurcation control of scalar smooth maps, Transactions of the ASME, Computational and Nonlinear Dynamics 15 (2020) 011001.
  * [14] M. Gazor, M. Kazemi, Normal form analysis of $\mathbb{Z}_{2}$-equivariant singularities, Int. J. Bifurcation and Chaos 29 (2019) 1950015–1950035.
  * [15] M. Gazor, M. Kazemi, A user guide for Singularity, arXiv preprint arXiv:1601.00268
  * [16] M. Gazor, M. Moazeni, Parametric normal forms for Bogdanov–Takens singularity; the generalized saddle-node case, _Discrete and Continuous Dynamical Systems_ 35 (2015) 205–224.
  * [17] M. Gazor, N. Sadri, Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms, _SIAM J. Applied Dynamical Systems_ 15 (2016) 870–903.
  * [18] M. Gazor, N. Sadri, Bifurcation controller designs for the generalized cusp plants of Bogdanov–Takens singularity with an application to ship control, SIAM J. Control and Optimization 57 (2019) 2122–2151.
  * [19] M. Gazor, A. Shoghi, Tone-colour in music and bifurcation control, J. Differential Equations 326 (2022) 129–163.
  * [20] M. Gazor, A. Shoghi, Bifurcation control and sound intensities in musical art, _J. Differential Equations_ 293 (2021) 86–110.
  * [21] M. Gazor, A. Shoghi, Leaf-normal form classification for n-tuple Hopf singularities, _Communications in Mathematical Physics_ 396 (2022) 481–526.
  * [22] M. Gazor, P. Yu, Spectral sequences and parametric normal forms, _J. Differential Equations_ 252 (2012) 1003–1031.
  * [23] G. Dangelmayr, J. Guckenheimer, On a four parameter family of planar vector fields,  Arch. Rational Mech. Anal. 97 (1987) 321–352.
  * [24] M. Golubitsky, I. Stewart, D.G. Schaeffer, _Singularities and Groups in Bifurcation Theory_ , Vol I and II, Springer, New York 1985 and 1988.
  * [25] B. Hamzi, Quadratic stabilization of nonlinear control systems with a double-zero bifurcation, _IFAC Proceedings_ 34 July (2001) 161–166.
  * [26] B. Hamzi, W. Kang, J.P. Barbot, Analysis and control of Hopf bifurcations, SIAM J. Control and Optimization 42 (2004) 2200–2220.
  * [27] B. Hamzi, W. Kang, A.J. Krener, The controlled center dynamics, SIAM J. Multiscale Modeling and Simulation 3 (2005) 838–852.
  * [28] B. Hamzi, J.S.W. Lamb, D. Lewis, A characterization of normal forms for control systems, J. Dynamics and Control Systems 21 (2015) 273–284.
  * [29] A.J. Homburg, B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, H. W. Broer, B. Hasselblatt, and F. Takens, eds., vol. 3, Elsevier Science, 2010, 379–524.
  * [30] W. Kang, Bifurcation and normal form of nonlinear control systems, PART I and II, SIAM J. Control and Optimization 36 (1998) 193–212 and 213–232.
  * [31] W. Kang, M. Xiao, I.A. Tall, Controllability and local accessibility: A normal form approach, IEEE Transaction on Automatic Control 48 (2003) 1724–1736.
  * [32] L.M. Perko, Differential Equations and Dynamical Systems, Springer, New York, 3rd edition, 2000.
  * [33] L.M. Perko, Multiple limit cycle bifurcation surfaces and global families of multiple limit cycles, J. Differential Equations 122 (1995) 89–113.
  * [34] H. Puebla, J.L. Ramirez, I. Cervantes, A simple tracking control for Chua’s circuit, _IEEE Transactions on Circuits and Systems, I. Fundamental Theory and Applications_ 50 (2003) 280–184.
  * [35] B.W. Qin, K.W. Chung, A. Algaba, A.J. Rodriguez-Luis, High-order approximation of heteroclinic bifurcations in truncated 2D-normal forms for the generic cases of Hopf-zero and nonresonant double Hopf singularities, SIAM J. Applied Dynamical Systems 20 (2021) 403–437.
  * [36] B.W. Qin, K.W. Chung, A. Algaba, A.J. Rodriguez-Luis, High-order analysis of global bifurcations in a codimension-three Takens-Bogdanov singularity in reversible systems, Int. J. Bifur. Chaos 30 (2020) 2050017, 18 pp.
  * [37] M.W. Rupniewski, W. Respondek, A classification of generic families of control-affine systems and their bifurcations, Math. Control Signals Systems 21 (2010) 303–336.
  * [38] D.D. Novaes, F.B. Silva, Higher order analysis on the existence of periodic solutions in continuous differential equations via degree theory, SIAM J. Math. Anal. 53 (2021) 2476–2490.
  * [39] J.J.E. Slotine, W. Li, _A_ pplied Nonlinear Control, Prentice-Hall, 1991.
  * [40] J. Yang, L. Zhao, Bifurcation analysis and chaos control of the modified Chua’s circuit system, _Chaos, Solitons and Fractals_ 77 (2015) 332–339.
  * [41] P. Yu, Closed form conditions of bifurcation points for general differential equation, Int. J. Bifurcation and Chaos 4 (2005) 1467–1483.
  * [42] Z. Wu, P. Yu, A method for stability and bifurcation control, IEEE Trans. Automat. Control 51 (2006) 1019–1023.
  * [43] H. Zhao, Y. Lin, Y. Dai, Hopf bifurcation and hidden attractor of a modifed Chua’s equation, _Nonlinear Dynam._ 90 (2017) 2013–2021.