RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Intermittency Chaos of Two Coupled Maps with Delay Coupling
Yuki Fujisawa† , Kenta Ago† , Yoko Uwate† and Yoshifumi Nishio††Departmaent of Electrical and Electronic Enginnering, Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506, Japan
Phone/FAX:+81-80-656-7000
E-mail: {fujisawa, kentago, uwate, nishio}@ee.tokushima-u.ac.jp

two maps with six periodic solution easily to become to thesynchronization states by delay.

Abstract

Generally, complex dynamical phenomena can be observed innetworks formed by many elements with nonlinearity. Cou2. Two coupled cubic maps
pled Map Lattice has proposed by Kaneko, to represent thecomplex high-dimensional dynamics, for example biologiA cubic map is expressed as follows:
cal systems, networks in DNA, and neural networks. In thisstudy, we investigate the influence of the delay in two coupledcubic maps with intermittency chaos. Moreover, the relationf (x) = ax3 + x(1 + a),
between average length of laminar part and the combination
where a represents a control parameter.
of delay is investigated.

(1)

1. Introduction
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 in DNA, economic activities and neural networks. Furthermore, we focus
on intermittency chaos and delay. The delay naturally occurs from information transmission and processing speeds inthe realistic networks[5]. In Ref.[5], the study investigated
the synchronization states of the coupled logistic maps withthe delay. As a result, the synchronization state of coupledchaotic maps are induced by the delay. Therefore, the studies considered the delay in coupled chaotic maps are investigated actively. In addition, intermittency chaos has stability
and mobility and gains good result 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 chaos. When we set
a control parameter of two cubic maps to generate intermittency chaos near the six periodic window, various synchronization states are confirmed in laminar part. Moreover, therelation between average length of laminar part and combination of the delay is investigated. Thereby, we could consider

aFigure 1: The bifurcation diagram of cubic map.

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 the boundary of periodic windows in the bifurcation diagram of cubic map. At
the boundary of six periodic window intermittency chaos isobserved in Fig. 2. Laminar represents the periodic state andburst represents the chaotic state. We define a = -3.69964153
representing the coexistence of laminar and burst in here. Figure 3 shows the time series of cubic map with intermittencychaos (a = -3.69964153). This figure shows intermittencychaos is switching between laminar and burst. Furthermore,in the case of a = -3.69964153, intermittency chaos including
six periodic laminars are observed. In this study, we consider
two coupled cubic maps with the delay.

- 459 -The coupling system is expressed as follows:

f(x)

f(x)

f(x)

the coupling strength, the difference of laminar and burst canbe clearly seen.
Next, we investigate the length of laminar part in coupled{
x(1,i+1) = (1 − g)f (x(1,i) ) + gf (x(2,i−τ1 ) )cubicmaps with the delay. In order to investigate quantitative(2)x(2,i+1) = (1 − g)f (x(2,i) ) + gf (x(1,i−τ2 ) ),average length of laminar part, we define laminar part by |x −±0.1895 or ± 0.4865 or ± 0.8873| < 0.0001. Figures 6 andwhere g represents the coupling strength, τ1 and τ2 represents 7 show the average length of laminar part during the iteration
the delay between the maps.
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 length of laminar parts arelonger. The other has become less than 200. Among the τ1 +
τ2 = 6n, than when the case of non-delay, combinations of τ1+ τ2 = 6n that the synchronization state is longer than nondelay is confirmed thirteen locations. From Fig. 7, if the caseof non-delay (τ1 =0, τ2 = 0), the average length of laminarpart is 10366, because the density of burst part is increased
when the coupling strength is high. In the case of τ1 + τ2 =6n, the average length of laminar parts are longer than Fig. 6.
The other 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 isaa
confirmed six locations.
Figure 2: The boundary of six periodic window.

if(x)(a) τ1 =0, τ2 =0.

iFigure 3: Time series of cubic map with intermittency chaos(a = -3.69964153).

i(b) τ1 =0, τ2 =3.

f(x)3. Simulation Results3.1 Average length of laminar partIn 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 is fixedwith i = 200000. And range of the delay is τ1 and τ2 = 0,1, · · ·, 12. Figures 4 and 5 show the time series of coupledcubic maps with the delay. Laminar during the iteration time
is longer than the other, e.g, Figs. 4 (a), (c) and Figs. 5 (a),(c). The other almost aren’t observed the laminar part, e.g,Figs. 4 (b) and Figs. 5 (b). As a result, increasing the value of- 460 -i(c) τ1 =1, τ2 =5.

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

f(x)

i

f(x)

(a) τ1 =0, τ2 =0.

Figure 7: Average length of laminar part (g = 0.00001).
ii(b) τ1 =0, τ2 =3.

f(x)3.2 Synchronization Patterni(c) τ1 =1, τ2 =5.

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

The synchronization states have six synchronization patterns in Fig. 8. Laminar part represents one of the synchronization states. There are all the patterns within each the timeseries. It does not have variation in the synchronization state
until switch to burst part. We investigate whether variation of
the synchronization state by the combination of the delay isobserved or not. In this study, we focus on the synchronization states of each of the length of laminar part is more than
3,000. All synchronization states of this laminar part in thetime series is the same synchronization pattern. If the case ofnon-delay, we confirm only in-phase synchronous, e.g, Fig. 8(a). However, Fig. 9 shows variation of the synchronization
states in the combination of τ1 + τ2 = 6n (n = 1, 2, 3, 4). In addition, we search the synchronization state that consists of inphase synchronous to anti-phase synchronous, it returns to thein-phase synchronous regularity. We have not 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 withthe delay. Next, we investigated the length of laminar partin coupled cubic maps with the delay. The average length oflaminar part is longer than the other when the delay is set toτ1 + τ2 = 6n (n = 1, 2, 3, 4). Thereby, we could consider twomaps with six periodic solution easily to become to the synchronization states when τ1 + τ2 = 6n. In addition to that wesearched the synchronization state consists of in-phase synchronous to anti-phase synchronous, it returns to the in-phasesynchronous regularity in the combination of τ1 + τ2 = 6n .

- 461 -f(x)

f(x)

f(x)

iiiii(d) Pattern 4.

(c) Pattern 3.

f(x)(b) Pattern 2.

f(x)

f(x)

(a) Pattern 1.

i(e) Pattern 5.

(f) Pattern 6.

Figure 8: Synchronization Patterns.
[4] K. Kaneko, “Simulating Physics with Coupled Map Lattice - Pattern Dynamics, Information Flow, and Thermodynamics of Spatiotemporal Chaos,” Formation, Dynamics, and Statistics of Patterns, World Sci., pp.1-52,
1990.
[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.
Figure 9: Regularity of the pattern.
[7] C. G. Langton, “Computation at the Edge of Chaos:Phase Transitions and Emergent Computation,” Physica
D, vol.42, pp.12-37, 1990.

References[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 SpatiotemporalChaos,” 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.

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