平成21年度電気関係学会四国支部連合大会

337

18-13

Synchronization Phenomena in
Globally Coupled ParametricallyForced Logistic Maps
Hironori KUMENO(Tokushima University)

Yoshifumi NISHIO(Tokushima University)

1. Introduction
Synchronization is one of the fundamental phenomena
in nature, and one of typical nonlinear phenomena. Therefore, studies on synchronization phenomena of coupled oscillators are extensively carried out in various fields.
In the past we have investigated effects of parametric
excitation for synchronization [1]. In this study, for additional investigation of the effect of parametric excitation onsynchronization, we focus on a globally coupled system ofone-dimensional map. A typical scheme for global couplingis given byNPεxi (t + 1) = (1 − ε)f [xi (t)] + N

f [xj (t)]j=1

Figure 1: One-parameter bifurcation diagram. Vertical axis:
xi (t). Horizontal axis: α2 . α1 = 3.8 and τ = 1.

(1)

i = 1, 2, · · ·, N,
where ε ∈ [0, 1] is the coupling intensity. The globallycoupled map is a scheme that an average number of all
the maps affect each of the map, and similar to the system that we have studied. Hence we investigate synchronization phenomena in a globally coupled system of onedimensional maps which are forced into periodic parameterchange. The one-dimensional map used in this study is alogistic map since the map can be described by a simple
discrete equation.
Firstly, we describe behaviors and bifurcations of theparametrically forced logistic map. Next, we investigate
synchronization phenomema in the globally coupled parametrically forced logistic maps.

2. Parametrically forced logistic map
A parametrically forced logistic map used in this studyis described as:xi (t + 1) = αf (t)xi (t)(1 − xi (t)),
andFigure 2: Return map of parametrically forced logistic map.
α1 = 3.8, α2 = 4.0 and τ = 1.

confirmed with increasing coupling intensity ε. Figure 3
shows examples of synchronization phenomena. In Fig. 3,
upper figures show return maps and lower figures showphase differences between the maps. First, when coupling
parameter ε is small, the maps are almost asynchronous.
With increasing coupling intensity ε, the synchronization
states are changed as: asynchronous → synchronizing oftwo of the three maps which generate periodic attractors→ synchronizing of two of the three maps which generatechaotic attractors (see Fig. 3(a)) → synchronizing of all themaps which generate chaotic attractors (see Fig. 3(b)).

(2)(α1 , n(τ1 + τ2 ) < t ≤ n(τ1 + τ2 ) + τ1
α2 , n(τ1 + τ2 ) + τ1 < t ≤ (n + 1)(τ1 + τ2 ) ,(n = 1, 2, ...)
(3)
where αf (t) is a term of the parametric force and timevarying. The parametric force operation can be describedas follows: in the time interval n(τ1 + τ2 ) < t ≤ n(τ1 +τ2 ) + τ1 , the system is driven by parameter α2 ; while in
the interval n(τ1 + τ2 ) + τ1 < t ≤ (n + 1)(τ1 + τ2 ), thesystem is driven by parameter α2 during the duration τ2 .
In this study, we assume τ1 = τ2 = τ for simplify. Oneparameter bifurcation diagram that α1 and τ are fixed andα2 is varying is shown in Fig. 1. Periodic, quasi-periodic
and chaotic attractors are confirmed. Figure 2 shows anexample of chaotic return maps at α2 = 4.0.
αf (t) =3. Synchronization
We carry out computer calculations for the globally coupled three parametrically forced logistic maps and investigate synchronization phenomena of the coupled maps. In
this case, various kinds of synchronization phenomena are(a)(b)

Figure 3: Return maps and phase differences. α1 = 3.8, α2 =4.0 and τ = 1. (a) ε = 0.32. (b) ε = 0.41.

4. Conclusions
In this study, we investigated synchronization phenomena of globally coupled three parametrically forced logisticmaps. By carrying out computer calculations for the system, we confirmed various kinds of synchronization phenomena. The synchronization states are changed with increasing coupling intensity as: asynchronous → synchronizing of two of the three maps → synchronizing of all themaps.
Reference[1] H. Kumeno and Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proceedings of International Symposium on Nonlinear Theory and its Applications (NOLTA 08), pp. 128131, Sep. 2008