2010 International Symposium on Nonlinear Theory and its Applications
NOLTA2010, Krakow, Poland, September 5-8, 2010

Synchronization Phenomena of Globally Coupled Logistic Mapswith Time-Varying Parameters
Hironori Kumeno† , Yoshifumi Nishio† and Daniele Fournier-Prunaret‡†Dept. of Electrical and Electronic Eng., Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN‡LATTIS-INSA, Universite de Toulouse
135 avenue de Rangueil, 31077 Toulouse, France
Email: { kumeno, nishio } @ee.tokushima-u.ac.jp, daniele.fournier@insa-toulouse.fr
Abstract—Synchronization phenomena in globallycoupled logistic maps whose parameters are forced into periodic varying are investigated when four and five maps arecoupled. Various synchronization phenomena are observedby choosing a coupling intensity in both cases. The observed synchronization phenomena are fall into five generalcategories, which are asynchronous, self-switching phenomenon of synchronization, coexistence phenomena ofsynchronization states, synchronization of the total numberof the coupled maps minus one map and synchronizationof all the maps.

scheme for global coupling is given by:xi (n + 1) = (1 − ε) f [xi (n)] + Nεi = 1, 2, · · ·, N,N∑f [x j (n)]j=1(1)

where ε ∈ [0, 1] is the coupling intensity. The globally coupled maps are a scheme such that an average number of all
the maps aﬀect each of the map. The one-dimensional map
used in this study is a logistic map, since the map can be
described by a simple discrete equation. Mathematically,
the logistic map is written asx(n + 1) = αx(n)(1 − x(n)).

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1. Introduction
Synchronization is one of the fundamental phenomena innature, and one of typical nonlinear phenomena. Therefore,studies on synchronization phenomena of coupled systems
are extensively carried out in various fields, physics [1], biology [2], engineering and so on. Parametric excitation circuit is one of resonant circuits, and it is important to investigate various nonlinear phenomena for future engineering
applications. In a simple oscillator including parametric
excitation, Ref. [3] reports that the almost periodic oscillation occurs if nonlinear inductor has saturation characteristic. Additionally the occurrence of chaos is referredin Refs. [4] and [5]. The network of chaotic elements
can be modeled by a system of coupled one-dimensional
maps. Behavior generated in coupled system of chaotic
one-dimensional map is investigated in Refs. [6]-[8]. In
particular, Coupled Map Lattice (CML) and Globally Coupled Map (GCM) are well known as mathematical models
in discrete-time system. The research into CML and GCM
is important for not only modeling of nonlinear systemsof multiple degree of freedom but also application to biological networks and engineering. In the past we have observed eﬀects of parametric excitation of coupled van der
Pol oscillators [9]. Previously, we investigate synchronization phenomena in three coupled logistic maps involving
parametric force [10]. In this study, we propose now to investigate synchronization phenomena generated by a larger
number of coupled maps with globally coupling. A typicalIn this study, we investigate synchronization phenomenain the globally coupled logistic maps whose parameters areforced into periodic varying when four or five maps arecoupled. The paper is organized as follows. In the next
section, we present the parametrically forced logistic map.
Synchronization phenomena observed in the globally coupled maps are in Section 3. The last section is devoted tothe conclusion.
2. Parametrically forced logistic map
A parametrically forced logistic map used in this study
is described as:x(n + 1) = α f (n)x(n)(1 − x(n)),

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and

α , for each even value of “n”
 1
α2 , for each odd value of “n”
α f (n) = , (4)(n = 0, 1, 2, ...)
where α f (n) is a term of the parametric force and timevarying. In this system, two kinds of parameters, α1 andα2 , are alternately replaced every update. Figure 1 shows
an example of a return map of the parametrically forced
logistic maps. For the original logistic map, two-periodic
solution is observed for α = 3.0, while, three-periodic solution is observed for α = 3.83. These two solutions are- 492 -periodic, whereas in the logistic map involving parametricforce, a solution is chaotic as shown in Fig. 1 when theparameters α1 and α2 are set 3.0 and 3.83, respectively.
Namely, chaotic solution can be observed in the combination of two parameters that generate two kinds of periodic
solutions. In the following, the parameter values are fixed
as α1 = 3.0 and α2 = 3.83.

Figure 2: Lyapunov exponents in globally coupled parametrically forced logistic maps for α1 = 3.0 and α2 = 3.83.
Horizontal axis: ε. Vertical axis: λ.
Figure 1: Return map of parametrically forced logistic map
for α1 = 3.0 and α2 = 3.83.

3. Synchronization
Synchronization phenomena generated in the coupled logistic map involving parametric force are investigated for
one control parameter ε which is coupling intensity whenfour and five maps are coupled. In the following computer calculations, the logistic map parameters are fixed as
α1 = 3.0 and α2 = 3.8.
3.1. Four maps caseIn this subsection, we consider the case of N = 4, namely
four parametrically forced logistic maps are coupled. Figure 2 shows maximum Lyapunov exponents, which calculates how complex trajectory of the solution is, for onecontrol parameter ε. Various synchronization phenomena
are observed for ε. Examples of synchronization phenomena observed in the four coupled maps are shown in Figs. 3
and 4.
First, the Lyapunov exponent is positive and all the maps
behave chaotic when ε is zero, namely all the maps are notcoupled. When ε is small, the maps are not synchronized.
Increasing ε, a self-switching phenomenon of synchronization on four coupled maps is observed as shown in Fig. 3
when ε is set around 0.045. The phenomenon is that threeamong the four maps are synchronized and the combination of the synchronized pair changes with time. Figure 3shows time series of diﬀerences of x(n) between two maps.
Areas where the amplitudes of the time series are small correspond to in-phase synchronization in the figure. In Fig. 3,firstly, map 1, map 3 and map 4 are synchronized. However, after a while, the synchronous state breaks up and map
1, map 2 and map 4 are synchronized. As seen above, thesynchronous states switch with time. More increasing ε,
the Lyapunov exponent becomes around zero or negative.
In some parts of ε between 0.05 and 0.08, there exist someFigure 3: Time series of diﬀerences between two maps
when a self-switching phenomenon of the synchronization
are observed. ε = 0.045, α1 = 3.0 and α2 = 3.83.

lines of Lyapunov exponent. It means that the synchronization state is multi stable. Coexistence phenomena ofvarious synchronization states are observed for the valuesof ε as shown in Figs. 4(a-1) and (a-2). Two pairs of twochaotic maps are synchronized in Fig 4(a-1). While, threeamong the four chaotic maps are synchronized in Fig 4(a2). These two synchronization state are observed for thesame parameters, although initial values are diﬀerent. Thesynchronization states depend on the initial values. In theregion of the multi stable state, other coexistence phenomena of synchronization states are confirmed as;- 493 -

• coexistence of three kinds of synchronization statesthat are synchronization of two among the four chaotic
maps, synchronization of two pairs of two periodic
maps and synchronization of three among the four periodic maps,• coexistence of two kinds of synchronization states thatare synchronization of two pairs of two periodic maps
and two pairs which are synchronization of two periodic maps and quasi-synchronization of two maps,• coexistence of two kinds of synchronization states thatare synchronization of two among the four chaotic
maps and synchronization of two pairs of two chaoticmaps.

synchronized and the combination of the synchronized pairchanges with time. Figure 6 shows time series of diﬀerences of x(n) between two maps. More increasing ε, theLyapunov exponent becomes around zero or negative. In
some parts of ε between 0.05 and 0.095, there exist somelines of Lyapunov exponent. Coexistence phenomena ofvarious synchronization states are observed in the parts ofε as shown in Figs. 7(a-1) and (a-2). Three among the fiveperiodic maps are synchronized in Fig 7(a-1). While, Threeamong the five periodic maps are synchronized and anothertwo periodic maps are synchronized in Fig 7(a-2). In the region of the multi stable state, other coexistence phenomenaof synchronization states are confirmed. It confirms the existence of some kinds of coexistence phenomena which areconstructed as combinations of following synchronization
states;

(a-1)

(a-2)

• quasi-synchronization of three among the five maps
and quasi-synchronization of two other maps,• quasi-synchronization of three among the five maps
and synchronization of two other maps,(b)• synchronization of three among the five maps and synchronization of two other maps,• two pairs of quasi-synchronization of two maps,• two pairs of synchronization of two maps,(c)• synchronization of two maps
synchronization of two maps,

3.2. Five maps caseIn this subsection, we consider the case of N = 5. Figure 5 shows maximum Lyapunov exponents for one-control
parameter ε. Various synchronization phenomena are observed for ε. Examples of synchronization phenomena observed in the five coupled maps are shown in Figs. 6 and7.
When ε is small, the maps are not synchronized. Increasing ε, a self-switching phenomenon of synchronization on
five coupled maps is observed as shown in Fig. 6 when εis set around 0.045. The phenomenon is that three among
the five maps are synchronized and another two maps arequasi-• two pairs of the synchronization of two maps andquasi-synchronization of one of the two pairs and another map,Figure 4: Synchronization of four maps. α1 = 3.0 and
α2 = 3.83. (a) ε = 0.050. (b) ε = 0.160. (c) ε = 0.200.

Next, more increasing ε, the Lyapunov exponent becomes positive again. Synchronization of three among thefour chaotic maps is observed when ε = 0.160 as shown in
Fig. 4(b). Finally, when ε is equal and over 0.200, all themaps are synchronized as shown in Fig. 4(c).

and• two pairs of the synchronization of two maps andquasi-synchronization of the two pairs,• quasi-synchronization of three among the five maps.
Next, more increasing ε, the Lyapunov exponent becomes positive again. Synchronization of four among thefive chaotic maps is observed when ε = 0.190 as shown inFig. 7(b). Finally, when ε is equal and over 0.210, all themaps are synchronized as shown in Fig. 7(c).
4. Conclusion
In this study, we have investigated synchronization phenomena in globally coupled logistic maps whose parameters are forced into periodic varying when four and fivemaps are coupled. Various synchronization phenomena canbe observed by choosing a coupling intensity in both thefour coupled maps case and the five coupled maps case.
The observed synchronization phenomena fall into fivegeneral categories, which are asynchronous, self-switching
phenomenon of synchronization, coexistence phenomena- 494 -(a-1)(a-2)
Figure 5: Lyapunov exponents in globally coupled parametrically forced logistic maps for α1 = 3.0 and α2 = 3.83.
Horizontal axis: ε. Vertical axis: λ.
(b)

(c)

Figure 6: Time series of diﬀerences between two maps
when a self-switching phenomenon of the synchronizationare observed. ε = 0.045, α1 = 3.0 and α2 = 3.83.

of synchronization states, synchronization of the total number of the coupled maps minus one map and synchronization of all the maps.
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Figure 7: Synchronization of five maps. α1 = 3.0 and α2 =3.83. (a) ε = 0.050. (b) ε = 0.190. (c) ε = 0.210.

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