平成 28 年度電気関係学会四国支部連合大会 講演論文集 (2016 徳島大学)
2016 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

1-20Synchronization Phenomena in a Ring of Cubic Maps
Containing Delay Coupling
Yuki FUJISAWA1

Yoko UWATE1
Yoshifumi NISHIO1( Tokushima University)
1

1. Introduction
Generally, complex dynamical phenomena can be observed in networks formed by many elements with nonlinearity. Coupled Map Lattice (CML) has been proposed byKaneko [1]-[4], to represent the complex high-dimensionaldynamics. We investigate the relation between average
length of laminar part and combination of the delay. And
we consider coupling cubic map with four periodic solution
easily to become to the synchronization states by delay.

Figure 1: The bifurcation diagram of cubic map.

2. Coupled Cubic Maps
A cubic map is expressed as following equation:f (x) = ax3 + x(1 + a),
where a represents a bifurcation parameter. We consider
coupled maps with delays. The coupling method is expressed as following equation:

Figure 2: Time series of coupled cubic maps
(τ1 = 0, τ2 = 0, τ3 = 0).

xn+1 (i) = (1 − g)f (xn (i))+ (g/2)(f (xn−τ (j)) + f (xn−τ (k))),where g represents the coupling strength, τ represents thedelay between the maps and n represents the iteration
time. Figure 1 shows the bifurcation diagram of cubic
map. This figure shows periodic windows (near a = -3.69,-3.83). We focus on the boundary of periodic windows in
the bifurcation diagram of cubic map. Figure 2 shows thetime series of coupled cubic map with intermittency chaos(a = 3.82842712). This figure shows intermittency chaos
is switching between laminar and burst. Furthermore, inthe case of a = 3.82842712, intermittency chaos including
four periodic laminars are observed.

(a) τ1 = 1, τ2 = 1, τ3 = 2.

3. Simulation results(b) τ1 = 2, τ2 = 4, τ3 = 2.
In this study, the initial conditions and the parametersof cubic maps are fixed with a = 3.82842712, x1 = 0.1, x2= 0.2, x3 = 0.3, g = 0.000001. The iteration time is fixedwith n = 200000. And range of the delay is τm = 0, 1, 2,3, 4(m = 1, 2, 3). Figure 3 shows the diﬀerence by combination of delay. When τm is set to 0 or 2 or 4, coupling
cubic map with four periodic solution easily to becometo the synchronization states by delay. Furthermore, thesynchronization pattern is changed by combination of delay.

Figure 3: Time series of cubic maps containing delay.

Reference[1] K. Kaneko, “Spatiotemporal Intermittency in Coupled
Map Lattice,” Prog. Theor. Phys., vol.75, no.5, pp.10331044, 1985.
[2] K. Kaneko, “Pattern Dynamics in Spatiotemporal
Chaos,” Physica D, vol.34, pp.1-41, 1989.
[3] K. Kaneko, “Spatiotemporal Chaos in One- and TwoDimensional Coupled Map Lattice,” Physica D,vol.37,pp.60-82, 1989.
[4] C. G. Langton, “Computation at the Edge of Chaos:Phase Transitions and Emergent Computation,” Physica
D, vol.42, pp.12-37, 1990.

4. Conclusions
In this study, we have investigated the influence of thedelay in coupled cubic maps with intermittency chaos.
First, we could consider maps with four periodic solution
easily to become to the synchronization states when τmis set to 0 or 2 or 4. Next, synchronization pattern ischanged when τm is set to 0 or 2 or 4.

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