IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2016

Research of Two Coupled Maps with Intermittency Chaos
Yuki Fujisawa, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,Tokushima University
Email: {fujisawa, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Generally, complex dynamical phenomena
can be observed in networks formed by many elementswith nonlinearity. Coupled Map Lattice has proposed byKaneko, to represent the complex high-dimensional dynamics, for example biological systems, networks in DNA,and neural networks. In this study, we investigate the influence of the delay in two coupled cubic maps with intermittency chaos. Moreover, the relation between average
length of laminar part and the combination of delay is investigated.

2. Two coupled cubic maps
A cubic map is expressed as follows:
f (x) = ax3 + x(1 + a),(1)

where a represents a control parameter.

1. Introduction

1 This study revised NCSP’16[8]aFigure 1: The bifurcation diagram of cubic map.

f(x)Generally, complex dynamical phenomena can be observed in networks formed by many elements with nonlinearity. Coupled Map Lattice (CML) has proposed by
Kaneko [1]-[4], to represent the complex high-dimensional
dynamics, for example biological systems, networks inDNA, economic activities and neural networks. Furthermore, we focus on intermittency chaos and delay. The
delay naturally occurs from information transmission andprocessing speeds in the realistic networks[5]. In Ref.[5],the study investigated the synchronization states of the coupled logistic maps with the delay. As a result, the synchronization state of coupled chaotic maps are induced by thedelay. Therefore, the studies considered the delay in coupled chaotic maps are investigated actively. In addition, intermittency chaos has stability and mobility and gains goodresult for information processing. We consider that intermittency chaos is related to various phenomena[6][7], e.g,
information processing of the brain. In order to make clear
the mechanism of such phenomena in various fields, unveiling the roles of intermittency chaos is very important.
In this study, we focus on the influence of the delay in
two coupled cubic maps with intermittency chaos1 . When
we set a control parameter of two cubic maps to generateintermittency chaos near the six periodic window, various
synchronization states are confirmed in laminar part. Moreover, the relation between average length of laminar partand combination of the delay is investigated.

aa

Figure 2: The boundary of six periodic window.
Figure 1 shows the bifurcation diagram of cubic map.
This figure shows period-doubling bifurcations and periodic windows (near a = -3.69, -3.83). We focus on theboundary of periodic windows in the bifurcation diagramof cubic map. At the boundary of six periodic windowintermittency chaos is observed in Fig. 2. Laminar represents the periodic state and burst represents the chaoticstate. We define a = -3.69964153 representing the coexistence of laminar and burst in here. Figure 3 shows- 44 -

=0, τ2 = 0), the average length of laminar part is 10366,
because the density of burst part is increased when the coupling strength is high. In the case of τ1 + τ2 = 6n, theaverage length of laminar parts are longer than Fig. 6. Theothers has become less than 30. Among the τ1 + τ2 = 6n,than when the case of non-delay, combinations of τ1 + τ2
= 6n that the synchronization state is longer than non-delay
is confirmed six locations.

f(x)

f(x)

the time series of cubic map with intermittency chaos (a= -3.69964153). This figure shows intermittency chaos isswitching between laminar and burst. Furthermore, in thecase of a = -3.69964153, intermittency chaos including six
periodic laminars are observed.

ii

Figure 3: Time series of cubic map with intermittencychaos (a = -3.69964153).

f(x)In this study, we consider two coupled cubic maps withthe delay. The coupling system is expressed as follows:{x(1,i+1) = (1 − g) f (x(1,i) ) + g f (x(2,i−τ1 ) )(2)
x(2,i+1) = (1 − g) f (x(2,i) ) + g f (x(1,i−τ2 ) ),(a) τ1 =0, τ2 =0.

where g represents the coupling strength, τ1 and τ2 represents the delay between the maps.

i3. Simulation Results(b) τ1 =0, τ2 =3.

In this study, the initial conditions and the parameters of
two cubic maps are fixed with x(1,0) = 0.1, x(2,0) = -0.2, g= 0.0000001, 0.00001, respectively. The iteration time isfixed with i = 200000. And range of the delay is τ1 andτ2 = 0, 1,· · ·, 12. Figures 4 and 5 show the time seriesof coupled cubic maps with the delay. Laminar during theiteration time is longer than the other, e.g, Figs. 4 (a), (c)and Figs. 5 (a), (c). The others almost aren’t observed thelaminar part, e.g, Figs. 4 (b) and Figs. 5 (b). As a result,increasing the value of the coupling strength, the diﬀerenceof laminar and burst can be clearly seen.
Next, we investigate the length of laminar part in coupledcubic maps with the delay. In order to investigate quantitative average length of laminar part, we define laminar partby |x − ±0.1895 or ± 0.4865 or ± 0.8873| < 0.0001. Figures 6 and 7 show the average length of laminar part during
the iteration time. From Fig. 6, if the case of non-delay(τ1 =0, τ2 = 0), average length of laminar part is 1547. In
the case of τ1 + τ2 = 6n (n = 1,2,3,4), the average lengthof laminar parts are longer. The others has become lessthan 200. Among the τ1 + τ2 = 6n, than when the case ofnon-delay, combinations of τ1 + τ2 = 6n that the synchronization state is longer than non-delay is confirmed thirteen locations. From Fig. 7, if the case of non-delay (τ1- 45 -f(x)3.1. Average length of laminar parti(c) τ1 =1, τ2 =5.

Figure 4: Time series of coupled cubic maps
(g = 0.0000001).

f(x)i(a) τ1 =0, τ2 =0.

f(x)Figure 7: Average length of laminar part (g = 0.00001).

3.2. Synchronization Patterniif(x)(b) τ1 =0, τ2 =3.

i(c) τ1 =1, τ2 =5.

Figure 5: Time series of coupled cubic maps (g = 0.00001).

Laminar part is composed of one of the six synchronization states. There is all the patterns within each the timeseries in Figs. 8. For example, Figs. 8(a) show in-phasesynchronous and Figs. 8(b) show anti-phase synchronous.
It don’t have variation in the synchronization state until
switched to burst part. We investigate whether the synchronization state that constitutes the laminar part there isvariation by the combination of the delay. In this study, wefocus on the synchronization states of each of the length oflaminar part is more than 3,000. All synchronization statesof this laminar part in the time series is the same synchronization pattern. When the case of non-delay, we search
in-phase synchronous laminar parts are only. But, Fig. 9shows variation of the synchronization states in the combination of τ1 + τ2 = 6n (n = 1,2,3,4).
In addition, we searched the synchronization state consists of in-phase synchronous to anti-phase synchronous, it
returns to the in-phase synchronous regularity when we set
τ1 + τ2 = 6n. For example, when we set τ1 + τ2 = 6n, thesynchronization state change Figs. 8(a), (c), (e), (b), (f), (d),(a). We haven’t searched variation to the coupling strength
in the synchronization states.
4. Conclusions

Figure 6: Average length of laminar part (g = 0.0000001).

In this study, we have investigated the influence of thedelay in two coupled cubic maps with intermittency chaos.
First, we observed the time series of coupled cubic maps
with the delay.
Next, we investigated the length of laminar part in coupled cubic maps with the delay. The average length of laminar part is longer than others when the delay is set to τ1+ τ2 = 6n (n = 1,2,3,4). As a result, we could consider
two maps with six periodic solution easily to become to thesynchronization states when τ1 + τ2 = 6n.
Finally, we focus on the synchronization pattern of eachof the length of laminar part is more than 3,000 when τ1 +τ2 = 6n. We searched the synchronization state consists ofin-phase synchronous to anti-phase synchronous, it returns- 46 -

f(x)

f(x)

to the in-phase synchronous regularity in the combinationof τ1 + τ2 = 6n .

ii(a) Pattern 1.

Figure 9: Regularity of the pattern.

(b) Pattern 2.

f(x)

f(x)

References[1] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattice,” Prog. Theor. Phys., vol.75, no.5,
pp.1033-1044, 1985.

ii(d) Pattern 4.

i(e) Pattern 5.

[3] K. Kaneko, “Spatiotemporal Chaos in One- andTwo- Dimensional Coupled Map Lattice,” Physica
D,vol.37, pp.60-82, 1989.
[4] K. Kaneko, “Simulating Physics with Coupled Map
Lattice - Pattern Dynamics, Information Flow, andThermodynamics of Spatiotemporal Chaos,” Formation, Dynamics, and Statistics of Patterns, World Sci.,
pp.1-52, 1990.

f(x)

f(x)

(c) Pattern 3.

[2] K. Kaneko, “Pattern Dynamics in SpatiotemporalChaos,” Physica D, vol.34, pp.1-41, 1989.

i(f) Pattern 6.

Figure 8: Synchronization Patterns.

[5] F. M. Atay and J. Jost, “Delay, Connection Topology,and Synchronization of Coupled Chaostic Maps,”
Phys. Rev. Lett. 92, 144101- , 2004.
[6] Y. Pomeau and P. Manneville, “Intermittent Transition to Turbulence in Dissipative Dynamical Systems,” Comm. Math. Phys., vol.74, pp.189-197,1980.
[7] C. G. Langton, “Computation at the Edge of Chaos:Phase Transitions and Emergent Computation,” Physica D, vol.42, pp.12-37, 1990.
[8] Yuki Fujisawa, Kenta Ago, Yoko Uwate, Yoshifumi
Nishio, ”Intermittency Chaos of Two Coupled Maps
with Delay Coupling,” Proceedings of RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP’16), pp. 459-462,
Mar. 2016.

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