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Elena Rybalova, Eckehard Schöll, Galina Strelkova

Controlling chimera and solitary states by additive

noise in networks of chaotic maps

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Rybalova, E., Schöll, E., & Strelkova, G. (2022). Controlling chimera and solitary states by additive noise in

networks of chaotic maps. In Journal of Difference Equations and Applications (pp. 1–22). Informa UK Limited.

https://doi.org/10.1080/10236198.2022.2118580.

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Controlling chimera and solitary states by additive noise in networks

of chaotic maps

Elena Rybalovaa, Eckehard Sch¨ollb,c,d, Galina Strelkovaa

aInstitute of Physics, Saratov State University, Saratov, 410012, Russia;

bInstitut f¨ur Theoretische Physik, Technische Universit¨at Berlin, 10623 Berlin, Germany;

cBernstein Center for Computational Neuroscience Berlin, 10115 Berlin, Germany;

dPotsdam Institute for Climate Impact Research, 14473 Potsdam, Germany

ARTICLE HISTORY

Compiled August 19, 2022

ABSTRACT

We study numerically the spatio-temporal dynamics of ring networks of coupled

discrete-time systems in the presence of additive noise. The robustness of chimera

states with respect to noise perturbations is explored for two ensembles in which

the individual elements are described by either logistic maps or Henon maps in the

chaotic regime. The influence of noise on the behavior of solitary states is investigated

for a ring of nonlocally coupled Lozi maps. Numerical simulations are performed for

a set of different noise realizations and random initial conditions to provide reliable

statistical data. The type of dynamics of the considered networks is quantified by

using a cross-correlation coefficient. We find that there is a finite and sufficiently

wide region with respect to the coupling strength and the noise intensity where the

probability of observing the chimeras and solitary states is high.

KEYWORDS

Network; nonlocal coupling; chimera states; solitary states; logistic map; Henon

map; Lozi map

1. Introduction

Random fluctuations are unavoidable in real-world systems. They can appear as intrin-

sic noise or may come as external random perturbations. Studying system responses

to noise is very important from a viewpoint of stability of operating modes, predicting

and controlling the functioning of a majority of life-significant systems, such as infras-

tructure systems, power grids, transport networks, communication and information

systems, biological, epidemiological, economic and social networks [4, 13, 14, 25, 38],

etc. Numerical studies have demonstrated that besides a degrading function, noise can

play a constructive role and gives rise to new dynamical behavior and induces novel

spatio-temporal patterns which do not exist in networks of coupled dynamical sys-

tems without noise [2, 6, 19, 25, 31, 43, 51, 73]. The noise influence can also effectively

control the operating modes and essentially improve certain properties of the system

dynamics [2, 7]. Recently, the interest in the study of noise impact in real-world sys-

tems has significantly increased with the discovery of special partial synchronization

patterns, such as chimera states [1, 10, 29, 45, 47, 49, 61, 63, 81] and solitary states

CONTACT E. R. Author. Email: rybalov[email protected]

[8, 27, 34, 66]. A chimera state was reported for the first time for a network of nonlocally

coupled identical phase oscillators [1, 29]. This peculiar spatio-temporal pattern repre-

sents an intermediate stage at a transition from complete coherence (synchronization)

to fully incoherent (spatio-temporal chaos) dynamics and denotes the coexistence of

spatially localized domains with coherent (synchronized) and incoherent (desynchro-

nized) dynamics of network oscillators. Recent theoretical and numerical studies have

demonstrated that chimera states can emerge in networks with various types of local

dynamics and a variety of topologies [1, 10, 29, 45, 48, 49, 58, 60, 67, 72, 76, 82]. These

partial synchronization patterns were also observed in a wide range of experimental

systems [18, 21, 28, 30, 37, 53, 59, 75, 79]. Besides, chimera states can be linked to var-

ious processes occurring in real-world systems, for example, in neuroscience [5, 35, 62],

in power-grids networks [39, 41, 78], in social systems [20, 24], etc.

Solitary states represent another important type of partial synchronization patterns

and are defined as network states for which single or several elements behave differ-

ently compared with neighboring elements which can behave coherently or can be

completely synchronized. As a rule, solitary nodes are typically randomly distributed

over the network space and their number increases when the coupling strength between

the network elements decreases. It was uncovered that the emergence of solitary states

is related to the appearance of bistable individual dynamics due to nonlocal interac-

tion [27, 34]. Recently, solitary states have been found in networks of the Kuramoto-

Sakaguchi models and the Kuramoto oscillators with inertia [8, 26, 27, 34, 80], the

discrete-time systems [55, 67, 68, 70], the FitzHugh–Nagumo systems [40, 54, 64, 65],

models of power grids [9, 22, 74], and even in experimental setups of coupled pendula

[28].

The robustness of chimera states with respect to noise was investigated in nonlocally

coupled networks of discrete-time maps with Gaussian and uniformly distributed pa-

rameters [36, 42] and in the presence of additive and multiplicative noise [12, 44, 56],

as well as in networks of continuous-time systems [32, 46, 69, 83]. In particular, it

has been demonstrated that noise can induce the appearance of novel chimera pat-

terns, such as coherence-resonance chimeras in a ring network of FitzHugh-Nagumo

neurons [69], or new spatio-temporal patterns, e.g., solitary states and solitary state

chimeras in a network of nonlocally coupled Henon maps [11, 57]. The dependence

of the chimera lifetime on the noise was studied in nonlocally coupled networks of

periodic and chaotic oscillators [32, 56, 71, 83]. It was shown that amplitude chimeras

are long-living transients and can be controlled by noise influence. The noise permits

to decrease the lifetime of amplitude chimeras in a ring network of Stuart-Landau

oscillators [32, 83] and to increase it to infinity in ensembles of nonlocally coupled

chaotic oscillators [56, 71]. Nevertheless, it is still not completely understood how the

probability of observing chimera states in networks of nonlocally coupled chaotic os-

cillators depends on the additive noise intensity, the coupling strength, and random

initial conditions.

In contrast to the analysis of chimera states in the presence of noise, the robustness of

solitary states with respect to noise has not been elucidated at all until very recently,

except [17], where it was shown that the presence of noise in a ring of nonlocally

coupled FitzHugh-Nagumo oscillators leads to a transition from solitary states to

patched synchrony. Thus, a more detailed study is needed to analyze the behavior

of solitary states in the presence of external random fluctuations.

In the present paper we investigate the influence of additive noise on chimera states

in networks of nonlocally coupled logistic and Henon maps and on solitary states in a

network of nonlocally coupled Lozi maps. We explore how the existence of these pe-

culiar partial synchronization patterns depends on the coupling strength and random

initial conditions in the presence of external noise. In order to provide good statistical

data we use different noise realizations for each of 50 different realizations of ran-

domly distributed initial conditions. The cross-correlation coefficient between network

elements is calculated to characterize the observed spatio-temporal structures and to

automatically detect and count chimera states and solitary nodes in the considered

networks both without and in the presence of noise.

2. Networks under study

2.1. Model equations

In this paper we investigate numerically the spatiotemporal dynamics of a network

which is represented by a ring of nonlocally coupled chaotic maps and is subjected to

additive noise. The mathematical model of the network, in its general form, reads:

xi(n+ 1) = Fi(n) + σ

i+R

j=i−R

[Fj(n)−Fi(n)] + Dξi(n),(1)

yi(n+ 1) = Gi(n),

where xi(n) and yi(n) are dynamical variables, i= 1,2,3, . . . , N,N= 1000 is the total

number of elements in the ensemble, ndenotes the discrete time. Functions Fi(n) and

Gi(n) are defined by the right-hand sides of the equations of the respective map (they

will be described below). The elements within the ring are coupled in a nonlocal way,

i.e., each ith element is linked with Rneighbors on the left and right, respectively.

Parameter Rdenotes the coupling range and σis the coupling strength between the

elements. We choose and fix R= 320 for all our numerical simulations. The last term

in the first equation of (1) corresponds to the influence of additive noise which is

uniformly distributed within the interval [−1,1], where ξi(n) is a noise generator, and

Dis the noise intensity. We have also studied the effect of additive Gaussian noise

sources on the network of coupled maps. However, the obtained results have shown a

qualitative similarity to the case of uniformly distributed noise.

As individual elements in the network (1), we have used the logistic map [15, 16], the

Henon [23] and the Lozi [33] maps. The logistic map is the simplest one-dimensional

map demonstrating chaotic dynamics with a nonhyperbolic attractor and multistabil-

ity [3]. The logistic map is given as follows:

xl(n+ 1) = Fl(n) = αlxl(n)(1 −xl(n)),(2)

where xl(t) is the dynamical variable, αlis the control (bifurcation) parameter. We

fix αl= 3.8 which corresponds to the chaotic dynamics of individual nodes when they

are uncoupled.

The two-dimensional Henon map is described by the following equations:

xH(n+ 1) = FH(n)=1−αH(xH(n))2+yH(n),(3)

yH(n+ 1) = GH(n) = βHxH(n),

where xH(n) and yH(n) are the dynamical variables, and αH>0 and βH>0 are the

control parameters. The Henon map under high-ratio compression (βH→0) reduces

to the logistic map. When the parameters are changed, the Henon map undergoes a

period-doubling bifurcation cascade which results in the emergence of a nonhyperbolic

chaotic attractor [3]. The Henon map is also characterized by the property of multista-

bility [3]. In our calculations, the parameters of the Henon map are fixed at αH= 1.4

and βH= 0.2 which corresponds to the chaotic dynamics in each uncoupled element

and prevents the trajectories from divergence to infinity when the maps are coupled

in a ring. It was found [45, 47, 55, 67] that when logistic maps or Henon maps are

nonlocally coupled within a ring, the transition from complete chaotic synchronization

to spatio-temporal chaos is accompanied by the appearance of amplitude and phase

chimera states when the coupling strength decreases.

The two-dimensional Lozi map is defined by the following system of equations:

xL(n+ 1) = FL(n) = 1 −αL|xL(n)|+yL(n)),(4)

yL(n+ 1) = GL(n) = βLxL(n),

where xL(n), yL(n) are the dynamical variables, and αL>0 and βL>0 are the

control parameters. In contrast to the two aforementioned maps, the Lozi map belongs

to the class of chaotic maps which can be obtained as the Poincar´e section of the

hyperbolic class of Lorenz-type attractors. The Lozi map exhibits a quasihyperbolic

chaotic attractor [50], and there is no multistability in the system phase space. We also

fix αL= 1.4 and βL= 0.3 which provides the chaotic behavior of the individual maps

in (1) without coupling. As was shown in Ref.[67], the ring of nonlocally coupled Lozi

maps demonstrates the transition from complete coherence to incoherence through

the emergence of solitary states when the coupling strength between the elements

decreases. The nonlocal coupling causes the birth of bistability in the dynamics of

individual Lozi maps. It was found that it is precisely the coexistence of two attracting

sets (bistability) in the phase space that is the mechanism for the emergence of solitary

nodes in this network [68].

Due to multistability of the considered networks not all of the initial conditions

provide chimera states or solitary states with the equal number of solitary nodes.

The spatio-temporal dynamics of each network is studied for 50 different realizations

of randomly distributed initial states of the dynamical variables (xi(0), yi(0)), each

having its own additive noise realization. The noise realizations used do not change

when the coupling strength and the noise intensity are varied.

2.2. Cross-correlation coefficient

In addition to the graphical representations for tracking the changes in the network

dynamics (snapshots and spatio-temporal diagrams), we calculate the cross-correlation

coefficient for each element of the network [77] and then plot the spatial profile (spatial

distribution) of its values along the ring. The cross-correlation coefficient is defined as

follows:

C1,i =h˜x1(n)˜xi(n)i

ph(˜x1(n))2ih(˜xi(n))2i, i = 2,3, . . . , N, (5)

where ˜x=x(n)−hx(n)i,hx(n)iis the averaging of the xvariable over time T= 50000

iterations. This measure shows the degree of correlation between the first element of the

ensemble and all the others, and accordingly illustrates the synchronous or correlated

behavior of the ring elements. The values of C1,i range from −1 to +1, where +1 relates

to full in-phase synchronization, and −1 corresponds to full anti-phase synchronization.

C1,i = 0 indicates the absence of correlation between the network elements. Calculation

of the cross-correlation coefficient enables one to detect automatically phase chimera

states and solitary states in the ensembles under study. It was found earlier [77] that

an incoherent cluster of a phase chimera is characterized by irregular switchings of

cross-correlation coefficient values between ≈ −1 and ≈+1, that indicates the anti-

phase nature of the oscillations of elements in time. In the case of solitary state mode,

the correlation coefficient takes the values ≈+1 and ≈ −0.9.

2.3. Network dynamics without noise

We first briefly overview the peculiar spatiotemporal structures (chimeras and solitary

states) in the networks of Henon maps and Lozi maps without additive noise. Figure

1 shows exemplary snapshots of the xivariables (Fig. 1,a), spatial distributions of

C1,i (Fig. 1,b), and spatio-temporal plots of xi(n) (Fig. 1,c) for the chimera state

in the ensemble of nonlocally coupled Henon maps (Fig. 1, left column) and for the

solitary state in the ensemble of nonlocally coupled Lozi maps (Fig. 1, right column).

A chimera state observed in the ring of nonlocally coupled logistic maps is similar to

that in the Henon map ring (Fig. 1, left column).

The chimera state chosen in our simulation and shown in Fig. 1,a-c,I represents the

coexistence of a phase chimera (two incoherent clusters with elements 25 < i < 55

and 652 <i<727), an amplitude chimera (the incoherent cluster includes elements

300 < i < 420), and a solitary state chimera (the incoherent cluster with nodes

877 <i<950). As can be seen from Fig. 1,b,I, the cross-correlation coefficient values

tend to 1 or −1 for the oscillators belonging to the coherent clusters, the clusters of

the phase chimera and of the solitary state chimera, while they lie within the interval

C1,i ∈[−1,−0.8] for the amplitude chimera cluster (or C1,i ∈[0.8,1] if the first element

relative to which the cross-correlation coefficients are calculated belongs to the ampli-

tude chimera cluster, for example i= 100). To automatically detect phase chimeras we

take into account the fact that there are clusters where the cross-correlation coefficient

changes from −1 to 1 several times (inside the incoherent cluster of phase chimeras),

and clusters in which C1,i approaches only either −1 or 1 for all the oscillators in the

coherent clusters. However, introducing the additive noise violates the ideal picture

that C1,i is confined to values 1 and −1. Therefore, we must define the confidence

intervals [−1,−0.8] and [0.8,1] instead of −1 and 1, respectively. For the same reason,

since it is rather difficult to automatically detect the amplitude chimera in a noisy

ensemble, we do not determine its presence. Here, we did not distinguish between the

phase chimeras and the solitary state chimeras, since they are similar in the spatial

distributions of C1,i.

In the case of solitary states (Fig. 1,a-c,II), the snapshot consists of a coherent

plateau (typical states) and solitary nodes evenly distributed over the entire ensemble

(Fig. 1,a,II). The cross-correlation coefficients for oscillators in the typical state are

equal to 1 (or −1 if the first oscillator is a solitary node), and for solitary nodes

C1,i ≈ −0.9 (or 0.9, if the first oscillator is a solitary node) (see Fig. 1,b,II). In

the presence of additive noise, the absolute value of the cross-correlation coefficient

I II

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

Figure 1. Examples of a combined chimera state in the Henon map ring at σ= 0.275 (panel I) and a solitary

state mode in the Lozi map ring at σ= 0.18 (panel II) without noise. Snapshots of the xivariables (a), spatial

distributions of the cross-correlation coefficient (b), and spatio-temporal diagrams of xi(n) (c). Parameters:

αH= 1.4, βH= 0.2, αL= 1.4, βL= 0.3, R= 320, D= 0, N= 1000

changes, as in the case of chimera states. If the first oscillator is in the typical state, the

C1,i values lie within the interval [−1,−0.7] for the solitary nodes. If the first oscillator

is a solitary node, the cross-correlation coefficient for the solitary nodes takes the values

from the interval [0.7,1]. We have used these values to automatically find the solitary

nodes.

3. Effect of additive noise on chimera states

In this section, we consider the effect of external additive noise on the spatio-temporal

dynamics in the networks of nonlocally coupled chaotic maps with a nonhyperbolic

chaotic attractor. As individual elements we choose the logistic map (Sec.3.1) and the

Henon map (Sec.3.2). These ensembles can demonstrate different types of chimera

states and we now explore their robustness to additive uniformly distributed noise

sources with different intensities.

3.1. Network of nonlocally coupled logistic maps

We start with considering the dynamics of the logistic map network (1) without noise

and with increasing coupling strength σbetween the elements. The initial conditions

for the dynamical variables xi(0) are chosen to be randomly distributed in the in-

terval [0,1). The changes in the spatio-temporal structures of the logistic map ring

are illustrated in Fig. 2 by snapshots, spatial distributions of cross-correlation coeffi-

cients and spatio-temporal diagrams. At a weak coupling strength (0 < σ < 0.15), the

network shows the regime of complete spatio-temporal incoherence, which turns into

spatial incoherence with periodic dynamics of the elements in time when the coupling

strength increases (Fig. 2,I). When 0.25 < σ < 0.4, we have some probability to ob-

tain either the coexistence of amplitude and phase chimeras (Fig. 2,II) or only phase

chimeras (Fig. 2,III). For some realizations of the initial conditions, only amplitude

I II III IV

(a)

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

(b)

(c)

Figure 2. Dynamics of the noise-free network of nonlocally coupled logistic maps for different values of the

coupling strength σ: 0.24 (panel I), 0.295 (panel II), 0.345 (panel III), 0.38 (panel IV). Snapshots of the xi

variables (a), spatial distributions of cross-correlation coefficient (b), and spatio-temporal diagrams of xi(n)(c).

Parameters: αl= 3.8, R= 320, D= 0, N= 1000

chimeras occur in the ensemble, which are formed on the snapshot with profile discon-

tinuities. As the coupling strength increases further, the incoherent clusters of phase

and amplitude chimeras become narrower, and the probability of emerging amplitude

chimeras decreases. It was analytically shown in Ref.[47] that chimera states in the

ring of nonlocally coupled logistic maps with the control parameter α= 3.8 occur

up to the critical coupling strength σ≈0.41 where the spatial wave profile has two

turning points with infinite slope. For larger values of σ, the spatial profile in the limit

N→ ∞ is continuous, while for lower values it breaks at these turning points into

two discontinuous branches, and the incoherent clusters are characterized by spatially

uncorrelated sequences of nodes alternating randomly between these two branches.

Due to the finite number of nodes in our simulations, chimera states are observed in

our studies only up to σ≈0.4. However, for some realizations of the initial conditions,

profiles with two discontinuities and without any chimeras appear at σ < 0.4 (See

Fig. 2,IV). These profiles show the discontinuities but not the two coexisting stable

branches, since they may be confined to a very small range of nodes, and their basins

of attraction may have quite different sizes, so that for random initial conditions only

one of the two branches may be predominantly reached by each node, and hence the

incoherent bistable profile typical of chimeras is not observed. At σ > 0.62, the logis-

tic map ensemble shifts to the regime of complete synchronization in space, which is

characterized by chaotic behavior of the elements in time[47].

Now we add a uniformly distributed noise source to the first equation of the logistic

map network (1) and analyze its effect on the network dynamics. Figure 3 presents a

two-parameter diagram of regimes which shows the probability Pof observing chimera

states depending on the coupling strength σand the noise intensity D. The quantity

Pdenotes the normalized number of initial realizations P=K/50, where Kis the

number of initial sets for which chimeras arise in the network. Qualitatively, one may

distinguish between the intervals of weak (D∈[0,0.005]) and strong noise (D > 0.005).

As follows from the diagram (Fig. 3), for weak noise (D < 0.005), there is still

a wide interval of the coupling strength σwhere the chimera states are observed

with a rather high probability (yellow and orange colors). With increasing noise, the

probability of chimeras near the right boundary of this interval even increases because

noise induces transitions between the basins of attraction of the two branches of the

D P

0.2 0.25 0.4 0.45

0.005

0.025

0.03

0.2

0.8

Figure 3. 2D diagram in the (σ,D) parameter plane showing the probability Pof observing chimera states in

the logistic map ring. Calculations were carried out for 50 different realizations of randomly distributed initial

conditions, each having its own noise realization. Parameters: αl= 3.8, R= 320, N= 1000

profile, and thus the incoherence clusters of the phase chimera in the vicinity of the

discontinuity points of the spatial profile become better visible. Figures 4,I-IV show

the dynamics of the logistic map network for weak noise intensity and at the same

values of σas in the noise-free case (Fig. 2). It is seen that the snapshots of the

noisy network dynamics (Fig. 4,I-III) are rather similar to those without noise, and

only minor noise-induced fluctuations are observed on the spatial profiles. Figure 4,IV

illustrates the case when in the presence of noise the incoherence clusters of the phase

chimera near the discontinuity points of the snapshot are broadened (compare Fig. 2,IV

without chimeras and Fig. 4,IV with the phase chimera with the incoherence cluster

895 <i<899). The reason for this is the noise-induced transitions between the two

basins of attraction of the two branches, which broadens the incoherent clusters. This

constitutes a constructive influence of noise.

In the case of strong noise influence, D > 0.005, the width of the chimera region is

reduced markedly with respect to σ(Fig. 3). The changes in the network dynamics in

the presence of strong noise are illustrated in Figs. 5,I-IV for the same values of σas

in the previous cases. When the coupling strength is very weak, the incoherent pro-

file does not undergo qualitative changes, only small fluctuations occur in the system

profile (compare Fig. 2,I and Fig. 5,I). At a weak coupling strength which is related

to the observation of chimera states in the noise-free system, the noise effect destroys

the whole spatial structure in the network (Fig. 5,II). Now a strongly incoherent snap-

shot of spatial profile corresponds to the network dynamics (Fig. 5,a,II). However,

the elements still oscillate periodically in time (Fig. 5,c,II) and the cross-correlation

coefficients take rather large values C1,i >0.89 (Fig. 5,b,II). With increasing coupling

strength, a phase chimera with noisy coherent clusters is clearly observed in the logis-

tic map ring (Fig. 5,III). Finally, when the coupling strength corresponds to a spatial

profile with a discontinuity or narrow clusters of the phase chimera, additive noise

smears out and ”smooths” the profile (Fig. 5,a,IV), which is especially apparent on

the distribution of cross-correlation coefficient values (Fig. 5,b,IV).

In summary, from Fig.3 it follows that there exists an optimum non-zero noise

I II III IV

(a)

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

(b)

(c)

Figure 4. Dynamics of the network of nonlocally coupled logistic maps in the presence of weak noise D=

0.0045 and for different values of the coupling strength σ: 0.24 (panel I), 0.295 (panel II), 0.345 (panel III),

0.38 (panel IV). Snapshots of the xivariables (a), spatial distributions of cross-correlation coefficient (b), and

spatio-temporal diagrams of xi(n) (c). Parameters: αl= 3.8, R= 320, N= 1000

I II III IV

(a)

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

0.3

0.9

C1,i

−1

xi(n)

100

1 10000.2

1.0

(b)

(c)

Figure 5. Dynamics of the network of nonlocally coupled logistic maps in the presence of strong noise

D= 0.02 and for different values of the coupling strength σ: 0.24 (panel I), 0.295 (panel II), 0.345 (panel III),

0.38 (panel IV). Snapshots of the xivarables (a), spatial distributions of cross-correlation coefficient (b), and

spatio-temporal diagrams of xi(n) (c). Parameters: αl= 3.8, R= 320, N= 1000

intensity D where the σ-interval of chimera observation is maximum. This counter-

intuitive constructive effect of noise is in some sense can be treated as a resonance-like

effect since the constructive and destructive action of noise are balanced for some

intermediate value of the noise strength.

3.2. Network of nonlocally coupled Henon maps

The ring of nonlocally coupled Henon maps can demonstrate phase and amplitude

chimeras within a certain range of the coupling strength [10, 67]. The introduction of

multiplicative noise into this ensemble can induce the appearance of a solitary state

chimera [57], which is characterized by the presence of an incoherent cluster containing

only solitary nodes. This type of chimera can also be observed in the Henon map

ensemble without external noise, but its basin of attraction is small and is unlikely

observed for random initial conditions.

As before, we first briefly consider the spatio-temporal dynamics of the noise-free

I II III IV

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

V VI VII VIII

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

Figure 6. Dynamics of the noise-free network of nonlocally coupled Henon maps for different values of the

coupling strength σ: 0.135 (panel I), 0.190 (panel II), 0.230 (panel III), 0.255 (panel IV), 0.290 (panel V), 0.380

(panel VI), 0.395 (panel VII), 0.405 (panel VIII). Snapshots of the xivarables (a), spatial distributions of

cross-correlation coefficient (b), and spatio-temporal diagrams of xi(n) (c). Parameters: αH= 1.4, βH= 0.2,

R= 320, D= 0, N= 1000

Henon map ring when the coupling strength increases. The initial conditions of the

dynamical variables for all the elements xi(0), yi(0) are randomly distributed in the

intervals xi(0) ∈[−0.5,0.5] and yi(0) ∈[−0.15,0.15]. Figure 6 illustrates the changes

in the spatio-temporal dynamics of the ring of nonlocally coupled Henon maps for

selected increasing values of σ. In each of the eight panels, the upper row (a) in-

cludes snapshots of the variables xi, the middle one (b) shows spatial distributions

of cross-correlation coefficient (5), and spatio-temporal diagrams of xi(n) are given

in the bottom row (c). In the case of weak coupling, a dynamical regime is observed

when the network elements behave incoherently in space and non-periodically in time

(Fig. 6,I). Amplitude and phase chimera states may appear in the network when σis

varied within the range [0.17,0.385]. This interval of nonlocal coupling strength was

established on the base of the analysis of 50 different initial distributions of the dynam-

ical variables. For some realizations of the initial conditions, chimeras are observed at

a rather weak coupling strength, and for a number of other realizations, at a stronger

one. To illustrate the dynamics of the Henon map ring, one realization of the initial

conditions was chosen, which leads to chimera states only at the coupling strengths

corresponding to Figs. 6,II,IV,V. Figure 6,II shows the coexistence of a phase chimera

(incoherent clusters with elements 1 < i < 131, 196 <i<428, 717 <i<1000) and an

amplitude chimera (the incoherent cluster includes elements 480 <i<630). As the

coupling strength grows, it is possible to observe the coexistence of a phase chimera

(two incoherent clusters: 315 <i<417 and 836 <i<1000) and a solitary states

chimera (the incoherent cluster consists of elements 216 <i<224) (Fig. 6,IV) or only

a phase chimera (two incoherent clusters: 24 <i<45, 583 <i<587) (Fig. 6,V). For

some initial distributions of the dynamical variables, the σ-interval of chimera state

existence can be interrupted, and in this case, the dynamics of the Henon map network

corresponds to a completely incoherent spatial profile with periodic oscillations of the

elements in time (Fig. 6,III). An increase in the coupling strength within the range

of chimera existence leads to a decrease in the probability of observing amplitude

chimeras and solitary state chimeras. At the same time, the width of the incoherent

clusters of the phase chimeras decreases. When σ > 0.36 for the chosen initial con-

ditions, the phase chimeras are not visible and one can observe an xiprofile with

discontinuity (Fig. 6,VI), where the discontinuity decreases smoothly (branches must

approach each other) with increasing coupling strength and, as a result, the profile

must represent a smooth function. This behavior is similar as for the logistic map,

and a similar analytical argument has been derived for the Henon map in [67]. How-

ever, at certain values of σfrom the range corresponding to the establishment of a

profile with discontinuity, smooth spatial profiles may appear, as shown in Fig. 6,VII,

which can be replaced again by discontinuous profiles as the coupling strength in-

creases (Fig. 6,VIII). For σ > 0.41, the profile discontinuity gradually decreases and,

ultimately, the system goes into a chaotic synchronization mode (complete synchro-

nization in space and non-periodic dynamics of elements in time). It should be noted

that the jumps from discontinuous profiles to smooth ones and vice versa observed

within the parameter range σ∈[0.365,0.410] are caused by the peculiarities of the

phase space structure and are not related to using different initial conditions. This

means that all the used initial condition do not lead to a gradual transition from the

discontinuous profiles to the smooth ones with increasing the coupling strength unlike

the case of the ring of nonlocally coupled logistic maps.

Now let us turn to the analysis of the effect of additive noise on chimera states in

the ring of nonlocally coupled Henon maps. Figure 7 demonstrates the dependence

of the probability of establishing chimera states from random initial distributions of

D P

0.15 0.2 0.4 0.45

0.02

0.08

0.1

0.2

0.8

Figure 7. 2D diagram in the (σ,D) parameter plane showing the probability Pof observing chimera states in

the Henon map ring. Calculations were carried out for 50 different realizations of randomly distributed initial

conditions, each having its own noise realization. Parameters: αH= 1.4, βH= 0.2, R= 320, N= 1000

the dynamical variables on the coupling strength σand the noise intensity D. As can

be seen, when the noise intensity is low, there is a high probability of chimera state

observation within a rather wide interval of the coupling strength. The maximum

probability Poccurs in the interval σ∈[0.3,0.35]. However, at D > 0.01, a second

maximum and a second region where the chimeras can be observed appear for strong

coupling. These two regions are separated by a blue area, which corresponds to a low

probability of observing chimeras. When D > 0.03, the region of chimera existence

in the weak coupling range disappears, and only the second region remains. From the

analysis of Fig.7 it can be concluded that the right boundary of the chimera region

even shifts towards a stronger coupling with increasing noise intensity. However, we

assume that this effect is rather similar to that observed in the logistic map ring. In

the noise-free case when σ∈[0.35,0.42], the probability of chimera state observation

vanishes but it can be increased by applying noise influence.

We now analyze the network behavior at a fixed noise intensity D= 0.025 and for

different values of the coupling strength (Fig. 8). As shown above, the boundary of

the chimera state region shifts to stronger coupling strength, chimera states are no

longer observed at weak coupling strength, and the ensemble passes into the regime

of complete spatial incoherence illustrated in Fig. 8,I. However, the cross-correlation

coefficient does not vanish, i.e., the network elements demonstrate nearly periodic dy-

namics in time (Fig. 8,b,I). With stronger coupling, a chimera state can appear in the

network even in the presence of noise with D= 0.025, but the width of the incoherent

domain is larger than in the noise-free case (Fig. 8,II). Upon further increase of σ,

the probability of observing chimera states decreases, and the network dynamics is

typically characterized by a spatial wave profile with fluctuations due to noise pertur-

bations (Fig. 8,III). For even larger values of the coupling strength, we again find a

region of chimera existence. In the neighborhood of the discontinuity narrow regions

of incoherence occur which correspond to incoherent clusters of the phase chimera

(Fig. 8,IV).

We now fix the coupling strength σ= 0.355 and increase the noise intensity. The

I II III IV

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

Figure 8. Snapshots of the xivariables (a), spatial distributions of cross-correlation coefficient (b), and

spatio-temporal diagrams of xi(n) (c) for the Henon map network for a fixed noise intensity D= 0.025 and for

different values of the coupling strength σ: 0.255 (panel I), 0.285 (panel II), 0.36 (panel III), and 0.38 (panel

IV). Other parameters: αH= 1.4, βH= 0.2, R= 320, N= 1000

I II III IV

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

Figure 9. Snapshots of the xivariables (a), spatial distributions of cross-correlation coefficient (b), and

spatio-temporal diagrams of xi(n) (c) for the Henon map network for a fixed coupling strength σ= 0.355 and

for different noise intensities D: 0 (panel I), 0.02 (panel II), 0.04 (panel III), 0.06 (panel IV). Other parameters:

αH= 1.4, βH= 0.2, R= 320, N= 1000

corresponding changes in the network dynamics are shown in Fig. 9. The chosen value

of the coupling strength corresponds to a snapshot with two discontinuities in the noise-

free case (Fig. 9,I), and the probability of observing a chimera state is extremely small

(see Fig. 7). When the noise intensity is low, a phase chimera with narrow incoherent

clusters appears (Fig. 9,II). Increasing the noise intensity leads to the disappearance of

discontinuities in the spatial profile, and, accordingly, to the absence of chimera states

(Fig. 9,III). This case corresponds to the region with a low probability of detecting

chimera states (Fig. 7). When the noise intensity increases, we enter into the second

region with a high probability of chimera existence, and the spatial profile contains

narrow clusters of incoherence of the phase chimera (Fig. 9,IV).

I II III IV

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

Figure 10. Dynamics of the noise-free network of nonlocally coupled Lozi maps for different values of the

coupling strength σ: 0.1 (panel I), 0.18 (panel II), 0.22 (panel III), 0.24 (panel IV). Snapshots of the xivariables

(a), spatial distributions of cross-correlation coefficient (b), and spatio-temporal diagrams of xi(n) (c). Other

parameters: αL= 1.4, βL= 0.3, R= 320, D= 0, N= 1000

4. Influence of additive noise on solitary states in the ring of nonlocally

coupled Lozi maps

In this section we investigate the robustness of solitary states in a ring of nonlocally

coupled Lozi maps with respect to additive noise and analyze their existence depending

on the noise intensity and the coupling strength. The initial conditions of all the

dynamical variables in the Lozi map ring are randomly distributed in the intervals

xi(0) ∈[−0.5,0.5] and yi(0) ∈[−0.5,0.5].

The ring of nonlocally coupled Lozi maps without noise can demonstrate solitary

states when the coupling strength σbetween the elements is varied within the range

[0,0.22]. Figure 10 shows how the noise-free network dynamics changes with increas-

ing coupling strength. As can be seen from Fig. 10,I, for weak coupling strength, an

incoherent mode of solitary nodes is observed in the network and is characterized by a

snapshot with an incoherent plateau. Increasing the coupling strength changes the spa-

tial profile which consists now of a coherent basis with solitary nodes (Fig. 10,II,III).

The number of solitary nodes gradually decreases as σgrows. Finally, for σ > 0.22, all

solitary nodes disappear, and the network dynamics corresponds to a coherent spatial

profile, while the elements behave irregularly in time (Fig. 10,IV). When the coupling

is sufficiently strong, the network elements become completely synchronized.

We now consider the influence of a uniformly distributed additive noise source on

the formation and evolution of solitary states in the Lozi map ring. The obtained

numerical results are summarized in a two-parameter diagram of regimes (Fig. 11)

where the number of solitary states is plotted in dependence of the coupling strength

σand the noise intensity D. The quantity Nsis calculated by averaging the normalized

number of solitary nodes Sover all 50 different random initial distributions, where S

is the number of solitary nodes obtained for each initial realization normalized by the

total number of nodes N= 1000. As can be seen, the distribution of S forms an arc-

shaped domain with a maximum at D≈0.08 and σ≈0.11. Qualitatively, one may

distinguish between the intervals of weak and strong noise. Weak noise corresponds

to D < 0.02 for which the number of solitary nodes remains almost unchanged in

the presence of additive noise. If 0.02 < D < 0.04, only the right boundary of the

region of solitary node existence changes in dependence on the coupling strength. The

DNs

0 0.05 0.25 0.3

0.05

0.15

0.20

0.05

0.3

0.35

Figure 11. The average normalized number of solitary nodes Nsobserved in the ring of nonlocally coupled

Lozi maps in the (σ, D) parameter plane. The calculations were carried out for 50 different realizations of

randomly distributed initial conditions of the dynamical variables and noise. Other parameters: αL= 1.4,

βL= 0.3, R= 320, N= 1000

number of solitary nodes is the largest for weak coupling strength, 0 < σ < 0.11, and

is gradually reduced as σbecomes stronger. When D > 0.04, the left boundary of

the solitary state region also changes (in the region of weak coupling). In this case

the additive noise is equivalent to the introduction of an inhomogeneity into the Lozi

map network. This leads to the destruction of the spatio-temporal structure and to a

strong inhomogeneity in the network when the cross-correlation coefficient vanishes.

When the noise intensity increases, the region of solitary node existence is smoothly

narrowing, while the boundaries lying both in the regions of weak and strong coupling

smoothly shift to the value of σ≈0.1. At D > 0.085, the solitary states no longer

exist in the Lozi map ring.

The evolution of the Lozi map dynamics is illustrated in Fig. 12 for a fixed noise

intensity D= 0.04 and with increasing coupling strength. It is seen that at a weak cou-

pling strength, the additive noise almost does not change the spatio-temporal structure

in the ring (compare Figs. 10,I and 12,I). At a stronger coupling, the coherent basis of

the spatial profile becomes noisy but the number of solitary nodes remains the same

(also compare Figs. 10,II and 12,II). If the value of σcorresponds to the existence of

a few solitary nodes only, then the noise intensity D= 0.04 is quite sufficient to re-

duce the basin of attraction of a solitary node attractor, and the solitary nodes are no

longer observed in the network (Fig. 12,III). A further increase in the coupling strength

does not cause the existing dynamical regime to qualitatively change, but leads to the

increase of the cross-correlation coefficient (compare Figs. 12,III and 12,IV).

We now fix the coupling strength at σ= 0.16 and analyze the changes in the Lozi

map ring dynamics as the noise intensity increases. The corresponding snapshots of

the xivariables (upper row), spatial distributions of cross-correlation coefficient val-

ues (middle row), and spatio-temporal diagrams (lower row) are shown in Fig. 13.

The value of σis chosen in such a way to observe a sufficiently large amount of soli-

tary nodes in the network (Fig. 13,I). As noted above, the introduction of a relatively

low noise intensity leads only to a noisy plateau of the spatial profile, while the soli-

tary node number does not change in any way as compared with the noise-free case

I II III IV

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

Figure 12. Snapshots of the xivariables (a), spatial distributions of cross-correlation coefficient (b), and

spatio-temporal diagrams of xi(n) (c) for the Lozi map network for a fixed noise intensity D= 0.04 and for

different values of the coupling strength σ: 0.1 (panel I), 0.18 (panel II), 0.22 (panel III), and 0.24 (panel IV).

Other parameters: αL= 1.4, βL= 0.3, R= 320, N= 1000

I II III IV

(a)

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

−0.7

1.1

C1,i

−1

xi(n)

100

1 1000

−0.7

1.1

(b)

(c)

Figure 13. Snapshots of the xivariables (a), spatial distributions of cross-correlation coefficient (b), and

spatio-temporal diagrams of xi(n) (c) for the Lozi map network for a fixed coupling strength σ= 0.16 and for

different noise intensities D: 0 (panel I), 0.04 (panel II), 0.065 (panel III), 0.075 (panel IV). Other parameters:

αL= 1.4, βL= 0.3, R= 320, N= 1000

(Fig. 13,II). When approaching the solitary state region with increasing noise intensity,

the number of solitary nodes is markedly reduced since some elements jump on the

attractor of typical states under the noise influence. Their cross-correlation coefficients

lie within the interval C1,i ∈[−0.7,0.7] (Fig. 13,III). Finally, when the noise intensity

is sufficiently high, the solitary state mode is destroyed and spatially incoherent dy-

namics is observed in the Lozi map ring (Fig. 13,IV). It should be noted that there are

differences between incoherent regimes which exist for weak and strong coupling. In

the first case the values of the cross-correlation coefficients of all the network elements

are close to zero, while C1,i >0.9 in the second case.

Let us return to Fig. 13,III and pay attention to the oscillators, the cross-correlation

coefficient values of which lie in the interval C1,i ∈[−0.7,0.7]. As mentioned above,

the solitary state regime is characterized by the bistable dynamics of the individ-

ual elements of the network. This means that there are two attracting sets in the

phase space, one of which corresponds to the typical states or coherent dynamics

(black dots in Fig. 14,c,I-III) and the other one – to the solitary nodes (red dots in

Fig. 14,c,I-III). The introduction of additive noise causes some of the network oscilla-

tors to switch from the solitary node attractor to the typical state attractor (Fig. 14,II,

green dots). However, their behavior is not completely synchronous with that of the

typical states and thus their cross-correlation coefficient takes the values from the in-

terval C1,i ∈[−0.7,0.7] (Fig. 14,b,II). For a larger noise intensity, the probability of

switching between the two coexisting metastable states increases, and one can observe

the solitary nodes with C1,i <−0.7 in the network. However, they can switch between

the typical state attractor and the solitary state one (blue dots in Fig. 14,c,III). Despite

the fact that these oscillators were taken into account when calculating the solitary

nodes, one can consider them as ”imaginary” solitary nodes which belong to the typi-

cal state attractor for most of the time, but their oscillations are not synchronous with

those of the coherent part C1,i <−0.7.

5. Conclusion

In this study we have demonstrated how additive uniformly distributed noise influ-

ences the emergence and existence of special partial synchronization patterns, such as

chimera states and solitary states, in networks of nonlocally coupled chaotic maps. We

have studied the robustness of chimera states with respect to noise using ring networks

of either logistic maps or Henon maps. It is known that these networks demonstrate

a transition from complete chaotic synchronization to complete incoherence through

the appearance of phase and amplitude chimeras. The effect of noise sources on soli-

tary states has been investigated in a ring of Lozi maps with nonlocal interaction.

The latter typically exhibits the existence of solitary states along the transition to

spatio-temporal chaos when the coupling strength decreases.

For both cases (chimeras and solitary states) we have investigated how regions of

existence of these patterns change when the nonlocal coupling strength σand the

additive noise intensity Dare varied. To this purpose we have calculated the cross-

correlation coefficients between network elements. Using this correlation measure we

can diagnoze the type of spatio-temporal dynamics of the considered networks, as well

as detect phase chimeras and solitary states. In order to provide reliable statistical

data we have performed numerical simulations for 50 sets of randomly distributed

initial conditions each possessing its own independent noise realizations. Furthermore,

we have illustrated the effects of noise on chimera and solitary states by snapshots,

I II III

(a)

(b)

(c)

Figure 14. Snapshots of the xivariables (a), spatial distributions of cross-correlation coefficient C1,i (b),

and phase portraits of all elements (c) of the Lozi map for different values of the coupling strength and the

noise intensity: D= 0, σ= 0.19 (panel I), D= 0.05, σ= 0.19(panel II), D= 0.65, σ= 0.17 (panel III). Black

points correspond to the typical states, red ones to the solitary nodes, green ones to the typical states with

C1,i ∈[−0.7,0.7], and blue ones to oscillators which switch between the typical state attractor and the solitary

state one. Other parameters: αL= 1.4, βL= 0.3, R= 320, N= 1000

spatial distributions of cross-correlation coefficient values and space-time plots.

We have computed two-parameter diagrams of regimes in the (σ, D) parameter

plane, which show the probability of observing chimera states in networks of coupled

logistic maps and Henon maps. It has been shown that in the case of the logistic map,

the phase chimeras can exist within a rather wide interval of the noise intensity and a

finite range of the coupling strength, and a high probability of observing chimera states

is detected for rather strong noise. In the case of the Henon map, it has been found

that there are two clearly distinguished regions in the (σ, D) parameter plane, within

which the phase chimera regimes persist even for rather large values of the noise inten-

sity. However, in both networks, already very weak noise can completely suppress the

chimera states in the case of weak and strong nonlocal coupling. Interestingly, we have

observed a counter-intuitive non-monotonic dependence of the chimera existence upon

noise intensity in both networks, which is reminiscent of the constructive influence of

noise known from coherence resonance.

To analyze the robustness of solitary states in a ring network of nonlocally coupled

Lozi maps with respect to additive noise we have plotted the normalized number of

solitary nodes in the (σ, D) parameter plane. The main result is that even weak noise

can sustain the solitary state regime in the range of weak nonlocal coupling. Stronger

noise causes the solitary nodes to disappear for any values of the coupling strength. It

can be assumed that this effect is related to noise-induced preference of attractors [52].

In the presence of noise, a small basin of attraction of solitary nodes becomes smeared

and thus most of the system elements go to a typical attractor with a larger basin. The

similar results were reported in Ref.[17] for the network of coupled FitzHugh-Nagumo

neurons in the presence additive noise.

Our findings provide a detailed understanding of the stability of chimera and solitary

states and open the possibility of controlling them using additive noise sources.

Disclosure statement

The authors declare that there is no potential conflict of interest.

Funding

E.R. and G.S. acknowledge financial support from the Russian Science Foundation

(project No. 20-12-00119) (numerical simulation of the dynamics of the Henon and

Lozi map networks). E.S. acknowledges financial support from the German Science

Foundation (DFG-Projektnummer 429685422 and 440145547) (numerical simulation

of the dynamics of the logistic map network).

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