Synchronization Phenomena in Coupled
Logistic Maps Involving Parametric ForceHironori Kumeno

Yoshifumi Nishio

Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima, 770-8506 JAPAN
Email: kumeno@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima, 770-8506 JAPAN
Email: nishio@ee.tokushima-u.ac.jp

Abstract— Synchronization phenomena in coupled logisticmaps whose parameters are forced into periodic varying areinvestigated through the use of Lyapunov exponents. Whenthree maps are coupled, various synchronization phenomena areobserved by choosing a coupling intensity. The synchronization phenomena fall into three general categories, which areasynchronous, synchronization of two among the three maps
and synchronization of all the maps. In particular, in thesynchronization of two of the three maps, solutions of maps
behave periodic, quasi-periodic and chaotic for several coupling
intensities.

I. I NTRODUCTION
Synchronization is one of the fundamental phenomena innature, and one of typical nonlinear phenomena. Therefore,studies on synchronization phenomena of coupled systems areextensively carried out in various fields, physics [1], biology[2], engineering and so on. Parametric excitation circuit is oneof resonant circuits, and it is important to investigate various
nonlinear phenomena for future engineering applications. Ina 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 referred in Refs. [4] and [5]. The networkof chaotic elements can be modeled by a system of coupled
one-dimensional maps. Behavior generated in coupled systemof chaotic one-dimentional map is investigated in Refs. [6][8] In particular, Coupled Map Lattice (CML) and GloballyCoupled 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 multiple degree of freedom
nonlinear systems but also application to biological networks
and engineering. In the past we have investigated effects ofparametric excitation of coupled van der Pol oscillators [9].
In this study, for more detailed investigation of the effectof parametric excitation on synchronization, we focus on a
globally coupled system of simple one-dimensional maps. A
typical scheme for global coupling is given byxi (t + 1) = (1 − ε)f [xi (t)] + Nεi = 1, 2, · · ·, N,N
j=1f [xj (t)](1)

where ε ∈ [0, 1] is the coupling intensity. The globally coupledmaps are a scheme that an average number of all the maps

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affect each of the map, and similar to the perfect couplingsystem of oscillators that we have studied using van der Poloscillators. The one-dimensional map used in this study is alogistic map, since the map can be described by a simple
discrete equation. Mathematically, the logistic map is writtenasx(t + 1) = αx(t)(1 − x(t)).
(2)
In this study, we investigate synchronization phenomenain the coupled logistic maps whose parameters are forced
into periodic varying. When three maps are coupled, various
synchronization phenomena are observed by choosing a coupling intensity. The synchronization phenomena fall into threegeneral categories, which are asynchronous, synchronizationof two of the three maps and synchronization of all themaps. In three state space corresponding to the three coupled
map, the synchronization phenomena are investigated throughspread of solutions, and classified according to behavior ofthe solutions. Thus, Lyapunov exponent that calculate thetemporal spreading of the solutions is available for investigate the synchronization. The synchronization is investigatedquantitatively through comparing three-dimensional Lyapunov
exponent with the synchronization.
II. PARAMETRICALLY FORCED LOGISTIC MAP
A parametrically forced logistic map used in this study isdescribed as:x(t + 1) = αf (t)x(t)(1 − x(t)),
and(3)⎧⎨ α1 , n(τ1 + τ2 ) < t ≤ n(τ1 + τ2 ) + τ1
α2 , n(τ1 + τ2 ) + τ1 < t ≤ (n + 1)(τ1 + τ2 ) ,αf (t) =⎩(n = 1, 2, ...)
(4)
where αf (t) is a term of the parametric force and timevarying. The parametric force operation can be described asfollows: in the time interval n(τ1 + τ2 ) < t ≤ n(τ1 + τ2 ) + τ1 ,the system is driven by parameter α1 during the duration τ1 ;while in the interval n(τ1 + τ2 ) + τ1 < t ≤ (n + 1)(τ1 + τ2 ),the system is driven by parameter α2 during the duration τ2 .
In this system, two maps which have two kinds of parameters
are replaced alternately by the number of updates. Then, a1368

when the parameters α1 and α2 are set 3.0 and 3.83. Namely,
chaotic solution can be observed in the combination of two
parameters that generate two kinds of periodic solutions.
III. S YNCHRONIZATION

Fig. 1. One-parameter bifurcation diagram (top) and the Lyapunov exponents(bottom) for α1 = 3.8 and τ = 1. Horizontal axis: α2 .

(a)

(b)

Fig. 2. Return maps of parametrically forced logistic maps for τ = 1. (a)α1 = 3.0 and α2 = 3.83. (b) α1 = 3.8 and α2 = 4.0.

parameter giving a periodic solution and a parameter giving
another periodic solution can be combined. Of course, othercombinations, for instance two parameters giving a periodic
solution and chaotic solution or two parameters giving two
kinds of chaotic solutions, are possible. In this study, weassume τ1 = τ2 = τ for simplicity.
The one-parameter bifurcation diagrams and the Lyapunov
exponents are obtained as shown in Fig. 1. From the figures,
observations of periodic and chaotic attractors are confirmed.
Figure 2 show some examples of the return maps of theparametrically 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 twosolutions are periodic, whereas in the logistic map involving
parametric force, a solution is chaotic as shown in Fig. 2(a)Synchronization phenomena generated in the coupled logistic map involving parametric force are investigated for one
control parameter ε which is coupling intensity when threemaps are coupled and τ = 1. Given that state spases of
the three maps are a three-dimensional phase spase, threedimensional Lyapunov exponent is defined because there exist
Lyapunov exponent whose the number of dimension is thesame as the number of dimension of the phase spase. The
three-dimensional Lyapunov exponents obtained at α1 = 3.8,
α2 = 4.0 and τ = 1 are shown in Fig. 3. In Fig. 3, λ1
is maximal Lyapunov exponent. λ2 is a Lyapunov exponent
whose direction is orthogonal to that of the λ1 . λ3 is a
Lyapunov exponent whose direction is orthogonal to that ofthe λ1 and that of the λ2 . Thus, λ1 calculates how complex
trafection of the solution is. λ1 + λ2 calculates planar spreadof the solutions. λ1 + λ2 + λ2 calculates spatial spread ofthe solutions. Figure 4 shows attractors in three-dimensional
phase space. Spatial spreads of the solutions are investigatedby using Fig. 3(b). In Fig. 3(b), as increasing ε, for ε > 0.19,
λ1 + λ2 + λ3 becomes negative, whereas λ1 + λ2 is positive.
Namely, the solutions narrow spatially, whereas the solutions
spread planarly. Though, more increasing ε, λ1 + λ2 suddenly
become negative. The solutions narrow planarly. However, forε ≥ 0.32, ε > 0.19, λ1 + λ2 + λ3 become positive again.
Figure 5 shows observed synchronization phenomena corresponding to Fig. 4. In Fig. 5, upper figures show the return
maps and lower figures show the phase differences between
the maps. For small value of the coupling intensity ε, λ1 ispositive and λ2 and λ3 is almost the same as λ1 . All themaps are chaotic and not synchronized as shown in Fig. 5(a).
For ε ≥ 0.17, λ3 is close to λ2 , two of the three maps aresynchronized as shown in Fig. 5(b). In the figure, x1 equal
to x2 . Thus, map 1 and map 2 are synchronized. However, inrelation of phase between maps 2 and map 3, concentration
distribution of the phase is not uniform state, and two areas
are high density. Thus, it seem that some kind of relation ofphase exists beteen the unsynchronized maps. For ε > 0.18,
λ3 becomes negative. All chaotic solutions gather in two areas
as shown in Fig. 5(c). In the figures of the phase differences inFig. 5(c), the areas where the solutions exist are symmetricalabout the axis of x2 (t) = x3 (t). That means, if x2 is in an areaaround x2 = 0.5, x3 is in an area around x3 = 0.8. While, if
x2 is in the area around x2 = 0.8, x3 is in the area aroundx3 = 0.5. For ε > 0.19, λ2 becomes negative. In Fig. 5(d), thephase difference between maps which are not synchronizedbecome like circle. For ε > 0.22, λ1 becomes zero. Thatmeans the solutions of all the maps are quasi-periodic. For
ε > 0.24, λ1 becomes negative. That means the solutions of
all the maps are periodic (see Fig. 5(e)). For ε > 0.32, λ1 ,λ2 and λ3 becomes positive. All the maps become chaotic
as shown in Fig. 5(f). Additionally, two of the three maps

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(a)

(b)

(c)

(d)

(e)

(f)(a)(g)Fig. 4. Attractors in three-dimensional phase space for α1 = 3.8, α2 = 4.0and τ = 1. (a) ε = 0.100. (b) ε = 0.180. (c) ε = 0.185. (d) ε = 0.205. (e)ε = 0.300. (f) ε = 0.320. (g) ε = 0.410.

ACKNOWLEDGMENT
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.
R EFERENCES(b)Fig. 3. Lyapunov exponents in globally coupled parametrically forced logisticmaps for α1 = 3.8, α2 = 4.0 and τ = 1. Horizontal axis: ε. (a)Each of λ.
(b)Sums of λ.

remain in the synchronization. In the figure, the solutions in
all the maps spread over solution space. Though, the shapesof the return maps are different from that of the uncoupled
map which are introduced in Fig. 2(b). For ε > 0.42, λ2 andλ3 becomes the same. All the map are synchronized as shownin Fig. 5(g). In the figure, all the maps are chaotic and havethe same shapes as a map of the uncoupled map.
IV. C ONCLUSION
In this study, we investigated synchronization phenomenain the coupled logistic maps which are forced into periodic
parameter varying. When three maps are coupled, various
synchronization phenomena are observed by choosing thecoupling intensity. The synchronization phenomena fall into
three general categories, which are asynchronous, synchronization of two among the three maps and synchronizationof all the maps. In particular, in the synchronization of twoof the three maps, solutions of maps behave periodic, quasiperiodic and chaotic for several coupling intensities. Moreover,the synchronization was investigated quantitatively throughcomparing three-dimensional Lyapunov exponent with thesynchronization.

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(a)

(b)

(c)

(d)

(e)(f)(g)Fig. 5. Return maps and phase differences for α1 = 3.8, α2 = 4.0 and τ = 1. (a) ε = 0.100. (b) ε = 0.180. (c) ε = 0.185. (d) ε = 0.205. (e) ε = 0.300.
(f) ε = 0.320. (g) ε = 0.410.

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