arX

iv:2

111.

0234

9v1

[nl

in.C

D]

3 N

ov 2

021

Chaos synchronization with coexisting global fields

O. Alvarez-Llamoza1 and M. G. Cosenza2, 3

1Grupo de Simulacion, Modelado, Analisis y Accesabilidad,

Universidad Catolica de Cuenca, Cuenca, Ecuador2School of Physical Sciences & Nanotechnology, Universidad Yachay Tech, Urcuquı, Ecuador

3Universidad de Los Andes, Merida, Venezuela.

We investigate the phenomenon of chaos synchronization in systems subject to coexisting au-tonomous and external global fields by employing a simple model of coupled maps. Two states ofchaos synchronization are found: (i) complete synchronization, where the maps synchronize amongthemselves and to the external field, and (ii) generalized or internal synchronization, where themaps synchronize among themselves but not to the external global field. We show that the stabilityconditions for both states can be achieved for a system of minimum size of two maps. We considerlocal maps possessing robust chaos and characterize the synchronization states on the space of pa-rameters of the system. The state of generalized synchronization of chaos arises even the drive andthe local maps have the same functional form. This behavior is similar to the process of spontaneousordering against an external field found in nonequilibrium systems.

I. INTRODUCTION

A global interaction in a system takes place when allits elements share a common influence or source of in-formation. Global interactions occur in many physi-cal, chemical, biological, social, and economic systems,such as parallel electric circuits, coupled oscillators [1, 2],charge density waves [3], Josephson junction arrays [4],multimode lasers [5], neural networks, evolution mod-els, ecological systems [6], social networks [7], economicexchange [8], mass media influence [9, 10], and cross-cultural interactions [11]. A variety of phenomena can oc-cur in systems subject to global interactions; for example,chaos synchronization, dynamical clustering, nontrivialcollective behavior, chaotic itineracy [12, 13], chimerastates [14, 15], quorum sensing [16]. These behaviorshave been investigated in arrays of globally coupled os-cillators in diverse experiments [17–21]. Global interac-tions can provide relevant descriptions in networks pos-sessing highly interconnected elements or long-range in-teractions. Systems with global and local interactionshave also been studied [22].A global interaction field may consist of an external

influence acting on all the elements of a system, as in adriven dynamical system [23]; or it may arise from theinteractions between the elements, such as a mean field[24], in which case we have an autonomous dynamicalsystem. In most situations, systems subject to eithertype of global field have been studied separately.In this article, we investigate the dynamics of systems

subject to the simultaneous influence of external and au-tonomous global interaction fields. As a simple modelfor such systems, we study a network of coupled mapswith coexisting external and autonomous global fields.Specifically, we focus on the important phenomenon ofchaos synchronization. With this aim, we consider localmap units possessing robust chaos dynamics. A chaoticattractor is robust if there exists a neighborhood in itsparameter space where windows of periodic orbits are ab-sent [25]. Robust chaos is an advantageous property in

applications that require reliable functioning in a chaoticregime, since the chaotic behavior cannot be destroyedby arbitrarily small perturbations of the parameters.In Section 2, we present the model for a system subject

to coexisting autonomous and external global fields. Wedefine two types of synchronization states for this systemin relation to the external field, complete synchronizationand generalized synchronization, and characterize themthrough statistical quantities. In Section 3, we carry outa stability analysis of the synchronization states and de-rive conditions for their stable behavior in terms of thesystem parameters. Section 4 contains applications ofthe model with coexisting global fields for maps exhibit-ing robust chaos. The states of chaos synchronization forthe system and their stability boundaries are character-ized on the space of parameters expressing the strengthof the coupling to the global fields. In particular, we showthat the state of generalized synchronization of chaos canappear even when the functional forms of the externalfield and the local maps are equal, a situation that doesnot occur for this family of maps if only one global fieldis present. This behavior represents a collective orderingof the system in a state alternative to that of the drivingfield. Conclusions are given in Section 5.

II. COUPLED MAP NETWORK WITH

AUTONOMOUS AND EXTERNAL GLOBAL

FIELDS

As a model of a system of chaotic oscillators with coex-isting autonomous and external global fields, we considera network of coupled maps in the form

yt+1 = g(yt), (1)

xit+1 = (1− ǫ1 − ǫ2)f(x

it) + ǫ1ht + ǫ2g(yt), (2)

ht =1

N

N∑

j=1

f(xjt ), (3)

2

where xit represents the states variable of element i (i =

1, 2, . . . , N) at discrete time t; N is the size of the system;f(xi

t) describes the local chaotic dynamics; yt+1 = g(yt)is an external global field that acts as a homogeneousdrive with independent chaotic dynamics; ht(x

jt ) is an au-

tonomous global field that corresponds to the mean fieldof the system; ǫ1 is a parameter measuring the strengthof the coupling of the elements to the mean field ht; andǫ2 expresses the intensity of the coupling to the externalfield. We assume a diffusive form of the coupling for bothfields.

A synchronization state for the system Eqs. (1)-(3) oc-curs when the N elements share the same state; that is,xit = x

jt , ∀i, j. Then, the mean field becomes ht = f(xi

t),∀i.

Two types of synchronization states can be definedin relation to the external global field g(yt): (i) com-plete synchronization, where the N elements in the sys-tem are synchronized among themselves and also to theexternal driving field; i. e., xi

t = xjt = yt, ∀i, j, or

ht = f(xit) = g(yt); and (ii) internal or generalized

synchronization, where the N elements get synchronizedamong themselves but not to the external global field; i.e., xi

t = xjt 6= yt, ∀i, j, or ht = f(xi

t) 6= g(yt).

To characterize the synchronization states of the sys-tem Eqs. (1)-(3) we calculate the asymptotic time aver-age 〈σ〉 (after discarding transients) of the instantaneousstandard deviation of the distribution of state variablesσt, defined as

σt =

[

1

N

N∑

i=1

(xit − xt)

2

]1/2

, (4)

where

xt =1

N

N∑

i=1

xit. (5)

Additionally, we calculate the asymptotic time aver-age 〈δ〉 (after discarding transients) of the instantaneousdifference

δt = |xt − yt| . (6)

Then, a complete synchronization state xit = x

jt = yt,

∀i, j, where the maps are synchronized to the externalglobal field, is characterized by the the values 〈σ〉 = 0 and〈δ〉 = 0. An internal or generalized synchronization state

xit = x

jt 6= yt, ∀i, j, where the maps are synchronized

among themselves but not the external field, correspondsto 〈σ〉 = 0 and 〈δ〉 6= 0.

In practice, we set the numerical condition 〈σ〉 < 10−7

and 〈δ〉 < 10−7 for the zero values of these statisti-cal quantities to characterize the above synchronizationstates.

III. STABILITY ANALYSIS OF

SYNCHRONIZED STATES

The system Eqs. (1)-(3) can be written in vector formas

xt+1 = Mf(xt), (7)

where xt and f(xt) are (N +1)-dimensional state vectorsexpressed as

xt =

ytx1t

x2t...

xNt

, f(xt) =

g(yt)f(x1

t )f(x2

t )...

f(xNt )

, (8)

and M is the (N + 1)× (N + 1) matrix

M = (1 − ǫ1 − ǫ2)I+1

NC, (9)

where I is the (N +1)× (N +1) identity matrix and C isthe (N+1)×(N+1) matrix that represents the couplingto the global fields, given by

C =

(ǫ1 + ǫ2)N 0 · · · 0ǫ2N ǫ1 · · · ǫ1...

. . ....

...ǫ2N ǫ1 · · · ǫ1

. (10)

The linear stability condition for synchronization canbe expressed in terms of the Lyapunov exponents for thesystem Eq. (7). This requires the knowledge of the (N +1) eigenvalues of matrix M, given by

µk = (1− ǫ1 − ǫ2) +1

Nck, k = 1, 2, . . . , N +1, (11)

where ck are the eigenvalues of matrix C, correspondingto c1 = (ǫ1 + ǫ2)N , c2 = ǫ1N , and ck = 0 for k > 2,which is (N − 1)–times degenerated.Then, the eigenvalues of matrix M are

µ1 = 1, (12)

µ2 = 1− ǫ2, (13)

µk = 1− ǫ1 − ǫ2, k > 2, (14)

(N − 1)− times degenerated.

The eigenvectors of the matrix M satisfying Muk =µkuk are also eigenvectors of the matrix C, and they aregiven by

u1 =

11...11

, u2 =

01...11

, uk =

0a1a2...

aN

, (15)

3

where the components ak of the eigenvectors uk, k > 2,

satisfy the condition∑N

k=1 ak = 0, since the eigenvectorsuk are orthogonal to the eigenvectors u1 and u2.The eigenvectors of the matrix M constitute a com-

plete basis where the state xt of the system Eq. (7)can be expressed as a linear combination. In particular,the complete synchronization state of xt is associated tothe eigenvector u1, while the generalized synchronizationstate is represented by the eigenvector u2.The (N + 1) Lyapunov exponents (Λ1,Λ2, . . . ,ΛN+1)

of the system Eq. (7) are defined as

(eΛ1 , eΛ2 , · · · eΛN+1) =

limT→∞

(

magnitude of eigenvalues of∣

∣

∣

∏T−1

t=0 J(xt)∣

∣

∣

)1/T

,

where J is the Jacobian matrix of the system Eq. (7),whose components are

Jij =

[

(1− ǫ1 − ǫ2)δij +1

Ncij

]

∂[f(xt)]i∂xj

, (16)

where cij are the ij-components of matrix C and [f(xt)]iis the i-component of vector f(xt). Then, we obtain

eΛk = limT→∞

∣

∣

∣

∣

∣

µTk

T−1∏

t=0

f ′(xkt )

∣

∣

∣

∣

∣

1/T

, k = 1, . . . , N + 1.

(17)where µk, k = 1, 2, . . . , N + 1, are the eigenvalues ofmatrix M. Substitution of the eigenvalues µk, gives theLyapunov exponents for the system Eq. (7),

Λ1 = λg, (18)

Λ2 = ln(1− ǫ2) + limT→∞

1

T

T−1∑

t=0

ln∣

∣f ′(x1t )∣

∣ , (19)

Λk = ln(1− ǫ1 − ǫ2) + limT→∞

1

T

T−1∑

t=0

ln∣

∣f ′(xkt )∣

∣ ,

k > 2, (20)

where λg is the Lyapunov exponent of the driven mapg(yt), which is positive since g(yt) is assumed chaotic.Note that, in general, the limit terms in Eqs. (19) and(20) depend on ǫ1 and ǫ2 since the iterates xi

t are ob-tained from the coupled system Eqs. (1)-(3). At syn-chronization, these terms are equal and we denote themby λf .The stability of the synchronized state is given by the

condition

eΛk =∣

∣µkeλk∣

∣ < 1, (21)

where λ1 = λg; λk = λf , for k > 1.Perturbations of the state xt along the homogeneous

eigenvector u1 = (1, 1, . . . , 1) do not affect the coherenceof the system; thus the stability condition correspond-ing to the eigenvalue µ1 is irrelevant for the complete

synchronized state. Then, condition Eq. (21) with thenext eigenvalue µ2 provides the range of parameter val-ues where the complete synchronized state xi

t = yt, ∀i, isstable; i. e.,

∣

∣(1− ǫ2)eλf∣

∣ < 1 ⇒ 1−1

eλf< ǫ2 < 1 +

1

eλf. (22)

Equivalently, complete synchronization takes place whenΛ2 < 0.On the other hand, the internal or generalized syn-

chronization state xt of the system is proportional to theeigenvector u2 = (1, 1, . . . , 0). The stability condition ofthis state is given by the next degenerate eigenvalue µk,k > 2; that is,

∣

∣(1− ǫ1 − ǫ2)eλf∣

∣ < 1

⇒ 1− ǫ2 −1

eλf< ǫ1 < 1− ǫ2 +

1

eλf. (23)

The condition for stable generalized synchronization canbe also be expressed as Λk < 0.Because of the eigenvalues Eqs. (12)-(15), the stabil-

ity conditions Eqs. (22) and (23) can be achieved for anysystem size N ≥ 2. Equations (22) and (23) describe theregions on the space of the coupling parameters (ǫ1, ǫ2)where complete and internal synchronization can respec-tively occur in the system with coexisting global fields,Eqs. (1)-(3).

IV. APPLICATIONS

We consider the system Eqs. (1)-(3) with local dynam-ics described by the tent map

f(xit) =

r

2

∣

∣1− 2 xit

∣

∣ , (24)

which exhibits robust chaos for r ∈ (1, 2] with xit ∈ [0, 1].

We fix the local parameter at the value r = 2 and assumethe external driving field equal to the local dynamics, i.e.,g = f .Figure 1(a) shows the statistical quantities 〈σ〉 and 〈δ〉

that characterize the collective synchronization states forthis system as functions of the coupling parameter ǫ2,with fixed ǫ1 = 0.2. System size is N = 5000. La-bels indicate the regions of the parameter ǫ2 where dif-ferent synchronization states take place: D (desynchro-nized state) where 〈σ〉 6= 0 and 〈δ〉 6= 0; SG (generalizedor internal synchronization) corresponding to 〈σ〉 = 0and 〈δ〉 6= 0; and SC (complete synchronization) char-acterized by 〈σ〉 = 0 and 〈δ〉 = 0. Figure 1(b) showsthe Lyapunov exponents Λ1, Λ2, and Λ3 as functions ofǫ2, with fixed ǫ1 = 0.2, for a system of minimum sizeN = 2, since the stability conditions for the synchronizedstates are satisfied for N ≥ 2. The Lyapunov exponentΛ1 = λg = ln 2 is positive. The transition of the expo-nent Λ3 from positive to negative values signals de onset

4

of stable generalized synchronization (GS), while Λ2 = 0indicates the boundary of the complete synchronizationstate (CS) that is stable for Λ2 < 0. The correspondingboundaries of the regions GS and CS coincide exactly inboth figures.

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1

〈σ〉

〈δ〉

(a)

D

GS

CS

ǫ2

−4

−3

−2

−1

0

1

0 0.2 0.4 0.6 0.8 1

Λ1

Λ3 Λ2

(b)

D GS CS

Λ

ǫ2

FIG. 1. (a) Statistical quantities 〈σ〉 (red line) and 〈δ〉 (blueline) as functions of ǫ2 for the system Eqs. (1)-(3) with localtent map Eq. (24) with r = 2, and external field g = f . Fixedǫ1 = 0.2, size N = 5000. For each value of ǫ2 both quan-tities are averaged over 20000 iterates after discarding 5000transients. (b) Lyapunov exponents Λ1 = λg (black line), Λ2

(blue line) and Λ3 (red line) as functions of ǫ2 for the systemEqs. (1)-(3) with minimum size N = 2 and g = f with fixedǫ1 = 0.2. For each value of ǫ2 the Lyapunov exponents werecalculated with 25000 iterations after discarding 5000 tran-sients. In this case, λf = λg = ln 2. Labels on both figuresindicate D: desynchronized or incoherent state; GS: general-ized or internal synchronization; CS: complete synchroniza-tion.

Figure 2 shows the collective synchronization states ofthe system Eqs. (1)-(3) with the local tent map and ex-ternal drive g = f with r = 2 on the space of parameters(ǫ1, ǫ2). The regions on this space where the different

states occur are indicated by labels. The boundaries ofthe synchronized states are calculated analytically fromconditions Eqs. (22) and (23). These boundaries coincidewith the criteria for the quantities 〈σ〉 and 〈δ〉 character-izing each state, as explained above.

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ǫ1

ǫ2

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

D

GS

CS

E

FIG. 2. Synchronization states for the system Eqs. (1)-(3)with local tent map and external drive g = f on the space ofparameters (ǫ1, ǫ2). Fixed parameters: r = 2, N = 5000. La-bels indicate the regions where these states can be found: D:desynchronization; GS: generalized synchronization; CS: com-plete synchronization; E: escape. The boundaries determinedanalytically with Eqs. (22) and (23) and by the quantities 〈σ〉and 〈δ〉 coincide exactly. The region labeled E corresponds tocoupling parameter values (ǫ1, ǫ2) for which the state variablesof the system escape to infinite. The boundary for region Eis given by the upper stability boundary of the generalizedsynchronization state given by Eq. (23).

To understand the nature of the collective behaviors,Fig. 3 shows the attractors corresponding to the differ-ent synchronization states for the reduced size systemEqs. (1)-(3), with the local tent map and external driveg = f . The desynchronized state (D) in Fig. 3(a) has allpositive Lyapunov exponents and shows no definite struc-ture. In the internal synchronization state (GS) withΛ1 > 0, Λ2 > 0, and Λ3 < 0, displayed in Fig. 3(b),the dynamics collapses onto an attractor lying on theplane x1

t = x2t . This plane constitutes the synchroniza-

tion manifold where x1t = x2

t = xt. Thus, the chaoticattractor on this plane represents a nontrivial functionalrelation, different from the identity, between xt and thedrive yt. In general, for the state of generalized synchro-nization with N > 2, a chaotic attractor arises betweenthe times series of the mean field xt and that of the drivesignal yt. The complete synchronized state (CS), pos-sessing Λ1 > 0, Λ2 < 0, Λ3 < 0, is characterized by theattractor lying along the diagonal line x1

t = x2t = yt, as

shown in Fig. 3(c). In this situation, xt = yt.The emergence of a chaotic attractor is a characteristic

feature of generalized synchronization in a drive-responsesystem when the drive function g is different from theresponse system function f . The generalized or internal

5

0

1 0

1

1

(a)

x1t

x2t

yt

0

1 0

1

1

(b)

x1t

x2t

yt

0

1 0

1

1

(c)

x1t

x2t

yt

FIG. 3. Attractors on the three-dimensional phase space ofthe reduced size system Eqs. (1)-(3) with the local tent mapand external drive g = f . Fixed parameters: r = 2, ǫ1 =0.2. (a) Desynchronized state (D), ǫ2 = 0.2. (b) Generalizedsynchronization (GS), ǫ2 = 0.4. (c) Complete synchronization(CS), ǫ2 = 0.6.

synchronization state does not arise if only the externaldrive g = f acts on the system of tent maps Eqs. (1)-(3), i.e., if ǫ1 = 0; nor can it appear with mean fieldcoupling alone. The emergence of the GS state in thissystem when g = f requires the coexistence of both, theautonomous global field and the external drive. Thus,we have a situation where the presence of an autonomousglobal interaction allows the synchronization of the mapsin a state alternative to that of the forcing external field.

As another example, we consider a local chaotic dy-namics given the logarithmic map

f(xit) = b+ ln

∣

∣xit

∣

∣ . (25)

This map is unbound and possesses robust chaos, withno windows of periodicity, for the parameter interval b ∈[−1, 1] [26].

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

D

ǫ1

ǫ2

(a)D

GS CS

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ǫ1

ǫ2

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

(b)D

GS

D

FIG. 4. Synchronization states for the system Eqs. (1)-(3)with local logarithmic map Eq. (25) and N = 5000 on thespace of parameters (ǫ1, ǫ2). (a) External drive function equalto the local dynamics, g = f = −0.7 + ln |x|. (b) Exter-nal drive different from the local map, with g = 0.5 + ln |x|and f = −0.7 + ln |x|. Labels indicate the states D: desyn-chronization; GS: generalized synchronization; CS: completesynchronization. The boundaries determined from conditionsEqs. (22) (blue line) and (23) (red lines) coincide with thoseobtained with the quantities 〈σ〉 and 〈δ〉.

Figure 4(a) shows the synchronization states of thesystem Eqs. (1)-(3) with the local chaotic map Eq. (25)and external drive g = f on the plane (ǫ1, ǫ2). Labelsindicate the regions where the different synchronizationstates take place. The boundaries of the synchronizedstates are calculated numerically from Eqs. (22) and (23).

6

The lower boundary of the GS state is calculated fromEq. (23) where λf 6= λg . For the upper boundary of theCS state, the local maps are already synchronized to thedrive yt and therefore λf = λg; thus Eq. (23) gives astraight line on the plane (ǫ1, ǫ2). Figure 4(b) shows thesynchronization states corresponding to an external driveg 6= f ; only generalized synchronization (GS) can occurin this case. There is no escape in either situation, sincethe map dynamics is unbounded.

V. CONCLUSIONS

We have studied a coupled map model for a systemsubject to coexisting autonomous and external globalfields. We have investigated the states of chaos synchro-nization in this system, consisting of (i) complete syn-chronization, where the maps synchronize among them-selves and to the external global field, and (ii) generalizedor internal synchronization, where the maps synchronizeamong themselves but not to the external field. The gen-eralized synchronization state can be described by the ap-pearance of a chaotic attractor between the time seriesof the mean field of the system and the external drivingfield.We have performed the stability analysis for both syn-

chronization states and found that the stability condi-tions can be achieved for a system of minimum size of

two maps subject to a common drive. The equivalenceof the dynamics for a minimum size system is a charac-teristic feature of systems with global interactions [27].

By considering local tent maps and logarithmic mapsthat possess robust chaos dynamics, we have focusedon the chaotic synchronization behavior of the system.We have characterized the synchronization states on thespace of the coupling parameters by using the stabilityconditions of these states as well as statistical quantities,with complete agreement in all cases. The emergence ofthe state of generalized synchronization of chaos, whenthe drive and the local maps have the same functionalform, requires the presence of both global fields. Thisbehavior is similar to the phenomenon of spontaneous or-dering against an external field found in nonequilibriumsystems [9].

Our results suggest that, in addition to chaos syn-chronization, other collective behaviors either observedor absent in a system with only one type of global in-teraction can be modified when both autonomous andexternal global fields are present.

ACKNOWLEDGMENT

This work was supported by ViceCancillerıa de Investi-gacion e Innovacion, Universidad Yachay Tech, Ecuador.

[1] Y. Kuramoto, Chemical Oscillations, Waves and Turbu-

lence (Springer, Berlin, 1984).[2] N. Nakagawa and Y. Kuramoto, Physica D 75, (1994)

74.[3] G. Gruner, Rev. Mod. Phys. 60, (1988) 1129.[4] K. Wiesenfeld and P. Hadley, Phys. Rev. Lett. 62, (1989)

1335.[5] K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy,

Phys. Rev. Lett. 65, (1990) 1749.[6] K. Kaneko and I. Tsuda, Complex Systems: Chaos and

Beyond (Springer, Berlin, 2001).[7] M. Newman, A. Barabasi, and D. J. Watts, The Structure

and Dynamics of Networks (Princeton University Press,Princeton, NJ, 2006).

[8] V. M. Yakovenko, in Encyclopedia of Complexity and

System Science, edited by R. A. Meyers (Springer, NewYork, 2009).

[9] J. C. Gonzalez-Avella, M. G. Cosenza, V. M. Eguiluz,and M. San Miguel, New J. Phys. 12, (2010) 013010.

[10] M. G. Cosenza, M. E. Gavidia, J. C. Gonzalez-Avella,PloS One 15(4), (2020) e0230923..

[11] J. C. Gonzalez-Avella, M. G. Cosenza, and M. SanMiguel, PLoS One 7, (2012) e51035.

[12] K. Kaneko, Physica D 41, (1990) 137.[13] S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette,

Emergence of Dynamical Order: Synchronization Phe-

nomena in Complex Systems (World Scientific, Singa-pore, 2004).

[14] G. C. Sethia and A. Sen, Phys. Rev. Lett. 112, (2014)144101.

[15] A. Yeldesbay, A. Pikovsky, and M. Rosenblum, Phys.Rev. Lett. 114, (2014) 144103.

[16] J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz,Proc. Natl. Acad. Sci. USA 101, (2004) 10955.

[17] W. Wang, I. Z. Kiss, and J. L. Hudson, Chaos 10, (2000)248.

[18] S. De Monte, F. d’Ovidio, S. Dano, and P. G. Sorensen,Proc. Natl. Acad. Sci. USA 104, (2007) 18377.

[19] A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K.Showalter, Science 323, (2009) 614.

[20] M. R. Tinsley, S. Nkomo, and S. Showalter, Nat. Phys.8, (2012) 662.

[21] I. Omelchenko and E. Scholl, Nat. Phys. 8, (2012) 658.[22] N. B. Ouchi and K. Kaneko, Chaos 10, (2000) 359.[23] M. G. Cosenza and A. Parravano, Phys. Rev. E 64,

(2001) 036224.[24] A. Pikovsky and M. Rosenblum, Chaos 25, (2015)

097616.[25] S. Banerjee, J. A. Yorke, and C. Grebogi, Phys. Rev.

Lett. 80, (1998) 3049.[26] T. Kawabe and Y. Kondo, Prog. Theor. Phys. 85, (1991)

759.[27] A. Parravano and M. G. Cosenza, Phys. Rev. E 58,

(1998) 1665.