# Chameleon attractors in turbulent flows

Tommaso Alberti  Francois Daviaud  Reik V. Donner  Berengere Dubrulle  Davide
Faranda  Valerio Lucarini

###### Abstract

Turbulent flows present rich dynamics originating from non-trivial energy
fluxes across scales, non-stationary forcings and geometrical constraints.
This complexity manifests in non-hyperbolic chaos, randomness, state-dependent
persistence and unpredictability. All these features have prevented a full
characterization of the underlying turbulent (stochastic) attractor, which
will be the key object to unpin this complexity. Here we propose a novel
formalism to trace the evolution of the structural characteristics of phase-
space trajectories across scales, providing a full characterization of the
attractor. We demonstrate that the properties of the dynamically invariant
objects depend on the scale we are focusing on. In the case of laboratory
experiments on fluids we observe the emergence of an intrinsic timescale,
solely determined by nonlinear interactions, controlling the geometrical and
topological properties of phase-space trajectories. Given the changing nature
of such attractors in time and scales we term them chameleon attractors.

###### keywords:

Attractors , Turbulence , Stochastic , Multiscale

††journal: Chaos, Solitons & Fractals

[inst1]organization=INAF-Istituto di Astrofisica e Planetologia Spaziali,
addressline=via del Fosso del Cavaliere, 100, city=Rome, postcode=00133,
country=Italy

[inst2]organization=CEA, IRAMIS, SPEC, CNRS URA 2464, SPHYNX, city=Gif-sur-
Yvette, postcode=91191, country=France

[inst3]organization=Department of Water, Environment, Construction and Safety,
Magdeburg–Stendal University of Applied Sciences,
addressline=Breitscheidstraße, 2, city=Magdeburg, postcode=39114,
country=Germany [inst4]organization=Research Department I – Earth System
Analysis, Potsdam Institute for Climate Impact Research (PIK) – Member of the
Leibniz Association, addressline=Telegrafenberg, A31, city=Potsdam,
postcode=14473, country=Germany

[inst5]organization=SPEC, CEA, CNRS, Universite Paris-Saclay, city=Gif-sur-
Yvette, postcode=91191, country=France

[inst6]organization=Laboratoire des Sciences du Climat et de l’Environnement,
CEA Saclay l’Orme des Merisiers, UMR 8212 CEA-CNRS-UVSQ, Universit Paris-
Saclay & IPSL, city=Gif-sur-Yvette, postcode=91191, country=France
[inst7]organization=London Mathematical Laboratory, addressline=Margravine
Gardens, 8, city=London, postcode=W6 8RH, country=UK
[inst8]organization=LMD/IPSL, Ecole Normale Superieure, PSL research
University, city=Paris, postcode=75005, country=France

[inst9]organization=Department of Mathematics and Statistics, University of
Reading, city=Reading, country=UK [inst10]organization=Centre for the
Mathematics of Planet Earth, University of Reading, city=Reading, country=UK

We introduce appropriate tools to investigate the local in time and space
properties of attractors

We show that a turbulent attractor is sensitive to the emergence of an
intrinsic timescale solely determined by nonlinear interactions

The attractor adapts its geometric and statistical properties dynamically in
time with respect to the intrinsic timescale, we call such attractor a
”chameleon” attractor

##  1 Introduction

The dynamics of turbulent flows is usually described via macroscopic dynamical
laws, the Navier-Stokes equations (NSE), derived from the mesoscopic Boltzmann
equation [1]. They accurately describe the dynamics of averaged quantities
over spatial scales much larger than the mean free path length of fluid
molecules [2]. Nevertheless, when considering large-scale turbulent flows even
the most powerful computer on Earth fails to accurately describe their
behavior at all relevant scales. Hence, there is a need to develop accurate
yet efficient parameterizations for describing the impact of the unresolved
scales of motions on those of interest [3, 4].

One way to approach the description of fluid flows containing a large number
of scales is via statistical analysis as in the multifractal formalism
developed by Parisi and Frisch [5]. It is based on characterising the high-
order statistics of the velocity field increments, thought to be
representative of fluctuations at different scales, via a set of scaling
exponents [6]. Contrasting the usual idea, coming from critical phenomena,
that only a countable set of scaling exponents are relevant for a complete
characterization of the statistical features of fluid flows [7, 8], Parisi and
Frisch [5] introduced an infinite hierarchy of exponents, each belonging to a
given fractal set. These exponents account for all possible (infinite)
rescaling symmetries of the NSE, describing the existence of singularities in
the energy cascade mechanism in turbulent flows [6, 9]. Since the development
of the multifractal theory experimental measurements of the velocity field in
fluids have proved to be compatible with this picture [10, 11, 12, 13, 14,
15]. However, this approach only provides global information on the scale-
dependent properties of fluids via the probability of occurrence of a given
scaling exponent. Moreover, a direct computation of the multifractal spectrum
from the NSE is not possible [16], although it would be helpful to explore the
local statistics of velocity field fluctuations [9].

A complementary approach to the high-order statistics is provided in the
framework of dissipative chaotic dynamical systems [17], exploiting the fact
that the concepts of turbulence and chaos are closely connected [18]. Indeed,
three-dimensional viscous fluids, as described via the NSE, conform to this
class of systems, being characterized by strange attractors, i.e., phase-space
states toward which the system evolves for a wide range of initial conditions
resulting from a series of bifurcations [18]. However, the search for an
attractor underlying turbulent flows has only proved partially successful so
far [19, 20, 21, 22]. Indeed, several studies have suggested that the observed
dynamical processes can be associated with the existence of non-hyperbolic
strange and possibly stochastic attractors having a dimensionality much lower
than the number of degrees of freedom of the system [23, 24]. Non-
hyperbolicity manifests itself with the fact that the attractor is
heterogeneous in terms of its local properties of persistence and
predictability [24, 25]. When considering numerical models, this has important
implications also in terms of error dynamics and efficiency of data
assimilation [26].

In this work, we use a laboratory experiment under high Reynolds number
turbulent conditions to explore the active number of degrees of freedom at
different scales. For this purpose we use a time-dependent parameter providing
information on the symmetries of the turbulent steady state, thus allowing us
to reconstruct the underlying attractor. By combining a decomposition method,
to detect scale-dependent components, with concepts from extreme value theory
(EVT), to sample local properties of attractors, we trace the evolution of the
geometrical and topological properties of these invariant objects across
scales for a symmetric and an asymmetric turbulent state. While the former is
characterized by a scale-invariant attractor, the latter is features an
attractor that is scale- and time-dependent, being sensitive to the emergence
of an intrinsic timescale solely determined by nonlinear interactions.
Furthermore, we also demonstrate that the symmetric turbulent steady state is
characterized by a simple phase-space topology and geometry, resembling that
of a noisy fixed point. Conversely, the asymmetric turbulent steady state
displays scale-dependent non-hyperbolic features, moving from a noisy fixed
point like structure at small scales (random attractor) towards a two-lobe
chaotic attractor at large scales. Thus, because the attractor adapts its
geometric and statistical properties dynamically in time with respect to the
intrinsic timescale, we call such attractor a chameleon attractor.

##  2 Data

Our data originate from a turbulent von Karman flow, obtained by stirring
rapidly water in a vertical cylinder of length $L=180$ mm and radius $R=100$
mm. As a result of the forcing turbulence develops that produces a back-
reaction onto the two stirring counter-rotating impellers measured through two
torque-meters located along their common axis. The resulting torques
$C_{1}(t)$ and $C_{2}(t)$ can be seen as large-scale quantities reflecting the
complex behaviour of the fluid [27]. Similarly, the instantaneous rotation
frequencies $f_{1}(t)$ and $f_{2}(t)$ of the two impellers provide a global
measure of the large-scale circulation that develops under the action of the
angular momentum flux [28, 29]. Although $f_{1}(t)$ and $f_{2}(t)$ provide a
1D (time-only) projection of the full 4D (space-time) dynamics of the
turbulent flow, they preserve intrinsic properties of the full turbulent
system such as intermittency, bi-stability and, for special forcing
conditions, a stochastic attractor. Such a situation is observed when $C_{1}$
and $C_{2}$ are constant [25]; as a result, the two frequencies $f_{1}(t)$ and
$f_{2}(t)$ fluctuate in time, with a typical mean frequency of $f_{0}\sim 7$
Hz [25]. The corresponding turbulent flow is then characterized by a Reynolds
number $Re=2\pi R^{2}f_{0}\nu^{-1}\sim 3\times 10^{5}$, significantly
exceeding the estimated critical Reynolds number for turbulence onset,
$Re_{T}\approx 3500$. However, the time fluctuations of $f_{1}(t)$ and
$f_{2}(t)$ follow an organized pattern determined by a control parameter
$\gamma=\langle(C_{1}(t)-C_{2}(t))/(C_{1}(t)+C_{2}(t))\rangle$ and traced by
an order parameter $\Theta(t)=(f_{1}(t)-f_{2}(t))/(f_{1}(t)+f_{2}(t))$. When
$\gamma=0$ the turbulent state is statistically symmetric and $\Theta(t)$
fluctuates around zero [27, 29]. For $\gamma\neq 0$, the symmetry is broken
and $\Theta(t)$ presents large-scale departures from zero.

Figure 1: The temporal behavior of a sample of $\Theta(t)$ for
$\gamma=-0.0081$ (red line) and $\gamma=0.0631$ (blue line). The horizontal
gray line refers to $\Theta(t)=0$.

Figure 1 reports the time behavior of a sample of $\Theta(t)$ for two selected
values of $\gamma$ used in this study,$\gamma=-0.0081$ (symmetric) and
$\gamma=0.0631$ (asymmetric). As expected, we note fluctuations around zero
for the symmetric case (i.e., $\gamma=-0.0081$), while large-scale transitions
featuring intermittent bursts are found for the asymmetric case (i.e.,
$\gamma=0.0631$). The difference between the two values of $\gamma$ can be
also highlighted by looking at the spectral properties of $\Theta(t)$ as
depicted by the variations of the power spectral density (PSD) across
frequencies as reported in Figure 2.

Figure 2: Power spectral density versus frequency for the two values
$\gamma\in\\{-0.0081,0.0631\\}$ as reported by red and blue lines,
respectively. The vertical dotted lines refer to the typical mean propeller
frequency $f_{0}=7$ Hz and its harmonic [25].

When $\gamma\sim 0$ the time series resembles that of an uncorrelated white
noise, typically characterized by a flat spectrum over a wide range of scales,
while for $\gamma>0$ a turbulent spectrum emerges. Furthermore, the
characteristic frequency $f_{0}\sim 7$ Hz associated with the average impeller
rotation frequency is also recognizable in the spectrum, together with its
harmonics at $2f_{0}$ [25]. Then, the spectrum saturates to that of a white
noise for $f>20$ Hz. To account for this behavior, in the following we apply a
low-pass filtering procedure with a cut-off frequency $f_{cut}\sim 20$ Hz to
our time series to reduce high-frequency fluctuations, also saving
computational time in our subsequent calculations [25].

##  3 Methods

###  3.1 Attractor reconstruction

As shown in Faranda et al. [25] the dynamical behavior of $\Theta(t)$ can be
globally described by a stochastic strange attractor whose geometry depends on
$\gamma$. Thus, as a first step of our procedure we reconstruct the global
attractor via Takens’ embedding method [19]. This means to translate our
univariate representation of the system in terms of the time series
$\Theta(t)$ into an $m-$dimensional manifold $\mathcal{M}$ via the following
diffeomorphism

$\Theta(t)\to\Theta_{m,\Delta}(t)=[\Theta(t),\Theta(t-\Delta),\Theta(t-2\Delta),\ldots,\Theta(t-(m-1)\Delta]^{\dagger}$
(1)

where $\dagger$ indicates the transposition operator. The two parameters,
i.e., the embedding dimension $m$ and the time delay $\Delta$, are selected
according to standard criteria based on the false nearest neighbor method,
suggesting $m=3$, and the time lag at which the auto-correlation function
reduces to 0.5, giving us $\Delta=20$ time steps [see, e.g., 25]. In this way,
we move from a univariate time series $\Theta(t)$ to a 3-D multivariate signal
$\Theta_{\mu}(t)=[\Theta_{1}(t),\Theta_{2}(t),\Theta_{3}(t)]$. The 3-D phase-
space for both values of $\gamma$ is reported in Figure 3.

Figure 3: 3-D reconstruction of the full attractor of the system for
$\gamma=-0.0081$ (red) and $\gamma=0.0631$ (blue).

A clear difference emerges between the two reconstructed attractors: while the
symmetric case ($\gamma\sim 0$) is characterized by a noisy fixed point like
structure, the asymmetric case is clearly characterized by a two-lobe
attractor, like that observed for many dissipative chaotic systems [17].
However, there is an additional complexity hidden in this global attractor,
that reflects the scale dependent properties of turbulence, as we demonstrate
in the following.

###  3.2 Multivariate Empirical Mode Decomposition (MEMD)

To uncover the scale dependence, we first decompose the data into intrinsic
modes by using the multivariate empirical mode decomposition [MEMD, 30] that
is the multivariate extension of the standard empirical mode decomposition
(EMD) [31]. It is an algorithmic procedure directly working in the data domain
to detect embedded patterns into multivariate signals $\Theta_{\mu}(t)$ in the
form of so-called Multivariate Intrinsic Mode Functions (MIMFs) [30]. These
patterns are derived through the sifting process [31], slightly modified to
implement an appropriate cubic spline procedure for multivariate signals [30].
It consists of the following steps:

  1. 1.

identify local extremes, i.e., points where the $\mu$-variate derivative of
$\Theta_{\mu}(t)$ is zero;

  2. 2.

use cubic spline interpolation over these points to derive the upper (maxima)
and the lower (minima) envelopes $\mathbf{U}_{\mu}(t)$ and
$\mathbf{L}_{\mu}(t)$, respectively;

  3. 3.

derive the mean envelope
$\mathbf{M}_{\mu}(t)=\frac{\mathbf{U}_{\mu}(t)+\mathbf{L}_{\mu}(t)}{2}$ and
evaluate the detail $\mathbf{H}_{\mu}(t)=\Theta_{\mu}(t)-\mathbf{M}_{\mu}(t)$.

These steps are iterated until the detail $\mathbf{H}_{\mu}(t)$ has the same
number of extrema and zeros (or having them differing at most by one) and a
zero-average mean envelope $\mathbf{M}_{\mu}(t)$. This means that
$\mathbf{H}_{\mu}(t)$ can be classified as the first Multivariate Intrinsic
Mode Function $\mathbf{C}_{\mu,1}(t)$ (also called multivariate empirical
mode) [31, 30]. Then, the algorithmic procedure is repeated over the first
residue $\mathbf{R}_{\mu,1}(t)=\Theta_{\mu}(t)-\mathbf{C}_{\mu,1}(t)$ until no
more MIMFs $\mathbf{C}_{\mu,k}(t)$ can be filtered out from the data, i.e.,
the final residue $\mathbf{R}_{\mu}(t)$ is a $\mu-$variate non-oscillating
(monotonic) trend [30]. Hence, we can write

$\Theta_{\mu}(t)=\sum_{k=1}^{N}{\bf C}_{\mu,k}(t)+{\bf R}_{\mu}(t).$ (2)

Each ${\bf C}_{\mu,k}(t)$ is a multivariate pattern representative of a
peculiar dynamical feature that evolves on a typical multivariate mean
timescale $\tau_{k}$ defined as [30]

$\tau_{k}=\frac{1}{N_{p}\,\Delta t}\int_{0}^{N_{p}\,\Delta
t}t^{\prime}\langle\mathbf{C}_{\mu,k}(t^{\prime})\rangle_{\mu}dt^{\prime},$
(3)

where $N_{p}$ is the number of data points, $\Delta t$ is the time resolution,
and $\langle\cdots\rangle$ stands for ensemble average over the
$\mu$-dimensional space. MIMFs are by construction ordered in terms of
decreasing frequency [30, 31]. Although an a priori decomposition basis is not
fixed, the derived basis, i.e., the set of $\\{{\bf C}_{\mu,k}(t)\\}$, is a
formal mathematical basis, that is, the MIMFs are empirically and locally
orthogonal with respect to each other [30]. Thus, partial sums of Eq. (2) can
be exploited to provide additional information over specific ranges of scales,
making the multivariate signal $\Theta_{\mu}(t)$ interpreted as a
superposition of scale-dependent fluctuations [32]. This property is used in
the following to diagnose the dynamical properties of the instantaneous (in
time) and local (in phase-space) states.

###  3.3 Dynamical system metrics

The dynamical properties of the $\mu-$variate systems can be investigated by
means of two dynamical systems metrics [33], the instantaneous dimension ($d$)
and the inverse persistence ($\theta$), based on extreme value theory (EVT).
The former is a measure of the active number of degrees of freedom, while the
latter is a measure of the short-term stability of the phase-space trajectory
associated with the extremal index of the generalized extreme value (GEV)
distribution of recurrence distances [34]. These instantaneous metrics are
obtained by sampling the recurrences (i.e., close encounters) of some
reference state $\zeta$ and observing that they are distributed according to
EVT [23, 33, 35].

Formally, let $x(\zeta)$ be the trajectory of the system and let
$\zeta^{\ast}$ be an arbitrary reference state in the phase-space. Let further
$g(x(\zeta),\zeta^{\ast})=-\log\left[dist(x(\zeta),\zeta)\right]$ be the
logarithmic return, where $dist(x(\zeta),\zeta^{\ast})$ is the Euclidean
distance between $x(\zeta)$ and $\zeta^{\ast}$. If we define exceedances as
$X(\zeta^{\ast})=g(x(\zeta),\zeta^{\ast})-s(q,\zeta^{\ast})$, with
$s(q,\zeta^{\ast})$ being an upper threshold corresponding to the $q$–th
empirical quantile of $g(x(\zeta),\zeta^{\ast})$, the Freitas-Freitas-Todd
theorem modified by Lucarini et al. (2014) [35] states that the cumulative
distribution $F(X,\zeta^{\ast})$ of returning to a sphere of radius $r$ around
$\zeta^{\ast}$ converges to the exponential member of the generalized Pareto
family

$F(X,\zeta^{\ast})\simeq\exp\left[-\theta(\zeta^{\ast})\frac{X(\zeta^{\ast})}{d^{-1}(\zeta^{\ast})}\right],$
(4)

where $0\leq d<\infty$ is the local dimension and $0\leq\theta\leq 1$ is the
inverse persistence of the state $\zeta^{\ast}$. Since each point of the
trajectory of the phase-space corresponds to a time instant of our embedded
time series by means of $d$ and $\theta$ we have a time-dependent view of the
properties of our system. However, this only provides information on the full
structure of the attractor, without revealing additional features that can be
related to processes and mechanisms operating at different scales. For this
reason, following Alberti et al. (2020) [32] we firstly use the MIMFs to
reconstruct the dynamics at different ranges of frequencies by exploiting
partial sums of Eq. (2)

${\bf\Theta}^{f}_{\mu}(t)=\sum_{k|f_{k}=1/\tau_{k}>f^{\ast}}{\bf
C}_{\mu,k}(t),$ (5)

giving us a description of the dynamical features at frequencies larger than
$f^{\ast}$. Starting from the largest frequency (i.e., $k=1$) and adding lower
and lower ones ($k=2,3,...,N)$ we can introduce for each frequency $f$ a
scale-dependent instantaneous dimension $D(t,f)$ and inverse persistence
$\theta(t,f)$ by diagnosing the dynamical properties of
${\bf\Theta}^{f}_{\mu}(t)$.

##  4 Results

Figures 4 and 5 show the instantaneous dynamical system metrics for (a) the
symmetric case with $\gamma=-0.0081$ and (b) a case with full symmetry
breaking at $\gamma=0.0631$, respectively.

Figure 4: The 3-D attractor of the system at different frequencies colored by,
respectively, the scale-dependent instantaneous dimension $D(t,f)$ (left
panels) and the scale-dependent inverse persistence $\theta(t,f)$ for the
symmetric case $\gamma=-0.0081$. Moving from top to bottom we consider high to
low cutoff frequencies. The color bar for the left panels is saturated between
2 and 4 for visual purposes.

Figure 5: Same as Figure 4 but for the case $\gamma=0.0631$.

In the symmetric case (Figure 4), we observe that the geometrical properties
of the attractor in phase-space are completely invariant with respect to
frequency, suggesting that the properties of the system do not depend on the
scale. The scale-dependent instantaneous dimension $D(t,f)$ slightly depends
on frequency, although the most probable and the average value $\langle
D(t,f)\rangle\approx 3$ are about the same for all frequencies. The scale-
dependent inverse persistence $\theta(t,f)$ is mostly characterized by values
larger than 0.8, with an average value $\langle\theta(t,f)\rangle\approx 1$
for all frequencies as expected for an unstructured stochastic system [25]. By
contrast, in the non-symmetric case (Figure 5), this scale-invariance is
broken. As a result, we observe sudden bursts of scale-dependent instantaneous
dimensions $D(t,f)\geq 6$, temporally localized differently at different
frequencies. The scale-dependent inverse persistence $\theta(t,f)$ displays
also markedly different behaviour across frequencies with respect to the
symmetric case ($\gamma\sim 0$), with values close to one at high frequencies
and less than 0.2 at lower ones. Furthermore, we clearly observe a transition
towards a two-lobe attractor at frequencies less than 0.2 Hz, thus matching
the expected phase-space geometry of the full attractor as well as the break
in the scaling observed in the PSD (see Figure 2). Thus, while in the
symmetric case, $\gamma=-0.0081$, we find that the $D(t,f)$ and $\theta(t,f)$
are homogeneously distributed across the attractor, implying that its topology
is very simple and compatible with a noisy fixed point, the non-symmetric case
attractor displays scale-dependent features with a heterogeneous spatial
distribution of the two metrics.

To further highlight the scale-dependent features, we show in Figure 6 the
behavior of the average dimension $\langle D(t,f)\rangle$ and persistence
$\langle\theta(t,f)\rangle$ in comparison with the PSD.

Figure 6: The behavior of the average dimension $\langle D(t,f)\rangle$ (upper
panel, filled circles) and persistence $\langle\theta(t,f)\rangle$ (lower
panel, filled circles) in comparison with the PSD (reported as solid lines).
Red symbols/lines refer to $\gamma=-0.0081$, while blue symbols/lines refer to
$\gamma=0.0631$.

We clearly observe that, on average, the symmetric case presents a scale-
invariant behavior of the two metrics, with values close to those expected for
a stochastic system (i.e., $\langle D(t,f)\rangle=3$ and
$\langle\theta(t,f)\rangle=1$). Conversely, a scale-dependent behavior is
observed in the non-symmetric case, with a transition between $\langle
D(t,f)\rangle<3$ and $\langle D(t,f)\rangle>3$ occurring around the low-
frequency break observed in the PSD. This also corresponds to changes from
$\langle\theta(t,f)\rangle<0.5$ to $\langle\theta(t,f)\rangle\to 1$. Thus, our
findings clearly suggest that we are faced with a scale-dependent modification
of the geometrical and topological properties of the underlying attractor,
depending on the emergence of an intrinsic timescale solely determined by
nonlinear interactions. This means that we observe a time behavior mirroring
the well-known scaling behavior of a 3-D turbulent flow. At scales larger than
the injection scale, the energy transfer is small, and the individual scales
are in quasi-equilibrium; for scales smaller than the injection scale, the
mean energy transfer is positive, and there is an out-of equilibrium energy
cascade towards smaller scales, following a Kolmogorov spectrum with
intermittency corrections [36]. In the present case, the low frequencies are
associated to low-dimensional dynamics, showing that the statistical
equilibrium at large scales is driven by a few degrees of freedom, generating
a well defined low-dimensional attractor. On the other hand, the dynamics at
scales smaller than the injection scale effectively plays the role of noise,
which restores the broken symmetry and provides the ”statistical temperature”
for large scales, or the stochasticity of the attractor [28]. Finally, since
our reconstructed 3D phase-space defined via $\Theta_{\mu}(t))$ is just a
projection of a higher-dimensional attractor where other degrees of freedom
are lump in stochastic terms (i.e., at small scales), it is not surprising
that we find dimensions larger than 3. As shown in Faranda et al. (2017) [25],
this points towards the existence of an unstable fixed point associated with
abrupt changes and hints at the existence of an underlying stochastic
attractor. Our scale-dependent results also suggest that, although the flow
dynamics involves a wide range of scales, some of them can be described by
stochastic theory [37].

##  5 Conclusions

In this paper we have introduced a new multiscale analysis tool to deal with
the investigation of time and scale-dependent properties of a simple system
derived from a turbulent flow. By exploiting two different cases with
different properties in terms of symmetry, we find evidence of a scale-
invariant nature of the geometrical properties of the phase-space for the
symmetric case. Conversely, in the non-symmetric case the scale-invariance is
broken by sudden bursts of elevated scale-dependent instantaneous dimensions
$D(t,f)$, temporally localized differently at different frequencies, with also
markedly different behavior of the local persistence properties. We clearly
observe a transition towards a two-lobe attractor at low frequencies matching
the expected phase-space geometry of the full attractor as well as the
frequency break observed in the spectral properties. Thus, we have
demonstrated that the geometrical and topological properties of invariant
objects (i.e., attractors) are in fact both scale and time-dependent, being
sensitive to the emergence of an intrinsic timescale solely determined by
nonlinear interactions. For this reason, since the studied attractor adapts
its geometric and statistical properties dynamically in time with respect to
the intrinsic timescale, we suggest to call such an attractor a chameleon
attractor. Furthermore, we also observed that the symmetric case has a very
simple phase-space topology and can be associated with a noisy fixed point,
while the non-symmetric case attractor displays scale-dependent features with
a heterogeneous spatial distribution.

Our results demonstrate that we cannot appropriately describe such attractors
with full/averaged properties, and that we need refined analysis tools to
detect their heterogeneity and the state-dependent properties of the system.
Hence, it is apparent that the analysis of multiscale systems requires
considering concepts allowing us to explore local and instantaneous properties
of the system [25, 37]. Our analysis shows that the highly heterogeneous
chameleon attractors discussed here could be common in high-dimensional
dynamical systems as those encountered in climate sciences. We are confident
that follow-up studies will further demonstrate their existence in such
systems by exploiting the framework applied in the present work.

## Acknowledgements

This work was funded through ANR EXPLOIT, grant agreement no.
ANR-16-CE06-0006-01 and ANR TILT grant agreement no. ANR-20-CE30-0035. VL
acknowledges the support received from the EPSRC project EP/T018178/1 and from
the EU Horizon 2020 project TiPES (Grant no. 820970). RVD has received funding
by the German Federal Ministry for Education and Research via the JPI
Climate/JPI Oceans project ROADMAP (grant no. 01LP2002B).

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