Delay-Induced Synchronization Statesin Two Coupled Cubic Maps
Yuki Fujisawa, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Email: {fujisawa, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— Generally, complex dynamical phenomena can beobserved in networks formed by many elements with nonlinearity.
Coupled Map Lattice has proposed by Kaneko, to represent
the complex high-dimensional dynamics, for example biologicalsystems, networks in DNA, and neural networks. In this study, weinvestigate 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 confirmedin laminar part. Moreover, the relation between average lengthof laminar part and the delay is investigated.

I. I NTRODUCTION

II. T WO COUPLED CUBIC MAPS
A cubic map is expressed as follows:
f (x) = ax3 + x(1 + a),
where a represents a control parameter.

(1)aFig. 1.

The bifurcation diagram of cubic map.

f(x)Generally, complex dynamical phenomena can be observedin 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 onintermittency chaos and delay. The delay naturally occurs
from information transmission and processing speeds in therealistic networks[5]. In Ref.[5], the synchronization states of
the coupled logistic maps are affected by the delay. Namely,
the synchronization state of coupled chaotic maps are induced by the delay. Therefore, the studies considered thedelay in coupled chaotic maps are investigated actively. In
addition, intermittency chaos has stability and mobility andgains good result for information processing. We consider thatintermittency chaos is related to various phenomena[6][7], e.g,
information processing of the brain. In order to make clear themechanism of such phenomena in various fields, unveiling theroles 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 confirmedin laminar part. Moreover, the relation between average lengthof laminar part and the delay is investigated.

iFig. 2. Time series of cubic map with intermittency chaos (a = -3.69964153).

Figure 1 shows the bifurcation diagram of cubic map.
This figure shows period-doubling bifurcations and periodic
windows (near a = -3.69,-3.83). At the boundary of periodic
windows intermittency chaos is observed. Intermittency chaos
is switching between laminar part and burst part. Laminar
represents the periodic state and burst represents the chaoticstate. Figure 2 shows the time series of cubic map withintermittency chaos. In this study, we consider two coupledcubic maps with the delay. The coupling system is expressed
as follows:x(1,i+1) = (1 − g)f (x(1,i) ) + gf (x(2,i−τ ) )x(2,i+1) = (1 − g)f (x(2,i) ) + gf (x(1,i−τ ) ),(2)where g represents the coupling strength, τ represents thedelay between the maps.

f(x){III. S YNCHRONIZATION S TATESIn this study, we focus on synchronization states of laminar part in two coupled maps. Additionally, laminar part iscomposed of some periodic patterns. In the case of a = 3.69964153, g = 0.0000001, intermittency chaos including six
periodic laminars are observed in Fig. 3. Here, the results
are considered that intermittency chaos including six periodic
laminar generates six patterns of synchronization states.

i

f(x)

f(x)

(d) i = 58970 ∼ 58990.

i(e) i = 7200 ∼ 7220.

i

f(x)

f(x)

(a) i = 6850 ∼ 6870.

i(f) i = 63350 ∼ 63370.

iFig. 3.
The pattern of synchronization state (a = -3.69964153, g =0.0000001).

(b) i = 3120 ∼ 3140.

IV. S IMULATION R ESULTS

f(x)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 and a = -3.69964153, respectively.
The iteration time is fixed with i = 200000. In the case of
a = -3.69964153, intermittency chaos including six periodic
laminars are observed in Figs. 4 and 5. Figures 4 and 5 shows
the time series of coupled cubic maps with the delay. FromFigs. 4 and 5, laminar part is longer than the others, when weset to τ = 3, 6.
i(c) i = 5000 ∼ 5020.

f(x)

f(x)

ii(a) τ = 0.

f(x)

f(x)

(a) τ = 0.

ii(b) τ = 1.

f(x)

f(x)

(b) τ = 1.

ii(c) τ = 2.

f(x)

f(x)

(c) τ = 2.

ii(d) τ = 3.
Fig. 4.

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

(d) τ = 3.
Fig. 5.

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

Average length of laminar part

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 part by |x −±0.1895 or ± 0.4865 or ± 0.8873| < 0.0001. Figure 6 shows
average length of laminar part during the iteration time. FromFig.6(a), when we set to τ = 9 and 27, the average length oflaminar part is longer than the case of non-delay (τ = 0). In
addition, when we set to τ = 3, 6 and 21, the average lengthof laminar part is similar length as much as the case of nondelay (τ = 0). From Fig.6(b), when the time delay is set to τ= 3, 6 and 9, the average length of laminar part is longer thanthe case of non-delay (τ = 0). Namely, the average length oflaminar part is longer than the others when the delay is set toa multiple of three.

delay

Average length of laminar part(a) g = 0.0000001.

delay(b) g = 0.00001.
Fig. 6.

Average length of laminar part (a = -3.69964153).

V. C ONCLUSIONSIn this study, we have investigated the influence of the delay
in two coupled cubic maps with intermittency chaos. First, weobserve the time series of coupled cubic maps with the delay.
Next, we investigate the length of laminar part in coupledcubic maps with the delay. The average length of laminar partis longer than others when the delay is set to a multiple ofthree. Thereby, we could consider two maps easily to becometo the synchronous states when delay’s number is the least
common multiple of periodic number.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.
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