2009 International Workshop on Nonlinear Maps and their Applications
NOMA'09, Urbino, Italy, September 10-11, 2009 (paper id: 012)

Investigation of Complicated Phenomenon inCoupled Cubic Maps
Yumiko Uchitani

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: uchitani@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization phenomena in coupled cubic maps. The cubic map has two attractors
located symmetrically with respect to the origin and they merge
as increasing a control parameter. By computer simulations,
we observe interesting state transition phenomenon as similar
to the case of the cross-coupled chaotic circuits. Further, wecompare the synchronization behaviors of the cross-coupledchaotic circuits and the coupled chaotic maps.

I. I NTRODUCTION
Synchronization phenomena in complex systems are very
good models to describe various higher-dimensional nonlinear
phenomena in the field of natural science. Studies on chaos
synchronization in coupled chaotic circuits are extensively
carried out in various fields [1]-[5]. We consider that it isvery important to investigate the phenomena related withchaos synchronization to realize future engineering application
utilizing chaos.
In our past studies [6]-[8], two Shinriki-Mori circuits [9][10]cross-coupled by inductors are investigated. As a result, wecould observe an interesting synchronization phenomenon, i.e.
the transitions from/to the positive region to/from the negative
region synchronize in in-phase or anti-phase. We consider that
this is caused by the symmetry of each circuit.
In this study, we investigate synchronization phenomena incoupled cubic maps. The cubic map has two attractors locatedsymmetrically with respect to the origin and they merge asincreasing a control parameter. Such bifurcations are similar
to those observed from Shinriki-Mori circuit. We compare
the synchronization behaviors of cross-coupled chaotic circuits
and the coupled chaotic maps.

II. C ROSS - COUPLED CHAOTIC CIRCUITS [6]-[8]In this section, we review the phenomena observed from thetwo cross-coupled chaotic circuits.
Figure 1 shows the cross-coupled chaotic circuits. In thismodel, two simple autonomous chaotic circuits called as
Shinriki-Mori circuit [9][10] are cross-coupled via inductors

Fig. 1.

Circuit model.

L2 . By using the following variables and the parameters,rr
L1 i1k
L1 i2k
xk =, wk =,

CVC

22 V

pv2kv1k yk = V , zk = V , t = L1 C2 τ,rr(1)
C2L1L1

α=,
β=G,
γ=g,

C1C2C2dL
1 δ=, “·”=L2dτ
the normalized circuit equations are given as follows where δ
is a coupling parameter.
ẋ1 = z1 ẋ2 = z2ẏ1 = α{γy1 − w1 − βf (y1 − z1 )} ẏ = α{γy − w − βf (y − z )}22222(2)ż=βf(y−z)
+w
−
x
1
1
121




 ż2 = β f (y2 − z2 ) + w1 − x2 ẇ1 = δ(y1 − z2 )ẇ2 = δ(y2 − z1 )where f are the nonlinear functions corresponding to the v −i characteristics of the nonlinear resistors consisting of the- 91 -

diodes and are assumed to be described by the following 3segment piecewise-linear functions:(yk − zk > 1)
 yk − z k − 10(|yk − zk | ≤ 1)
(3)
f (yk − zk ) =yk − z k + 1(yk − zk < −1)
From the circuit in Fig. 1, we could observe interestingstate transition phenomenon. Typical examples of the observed
phenomena are shown in Figs. 2 and 3. By computer simulations and circuit experiments, we confirmed that the circuitsgenerated many different synchronization states, by changing
initial condition.
2.5(a)(b)0-2.02.000y2-2.002.5

τ02.5

Two examples of different synchronization states. (circuitexperimental results). L1 = 10.56mH, L2 = 1.28H, C1 =33.3nF,
C2 =49.5nF and g=515mS. (a) Attractor on v11 −v12 plane. Horizontal
and vertical: 5 V/div. (b) Time waveform v11 and v21 . Horizontal 0.5
ms/div and vertical: 5 V/div.
Fig. 3.

500

2.0

y10-2.0
2.000y2-2.5(b)2.0y1-2.5(a)

0600

-2.0

2.5

τ

0500

500

2.5

2.0y1y20(a)2.5

tau0-2.5

4000-2.0
2.00-2.0

τ

0500

300

200(b)
100Fig. 2. Some examples of different synchronization states. (computersimulation results). α = 2.0, β = 4.0, γ = 0.1, and δ = 0.0014.
(a) Attractor on y1 − y2 plane. (b) Timewaveform.

Figure 4 shows how the sojourn times between the statetransitions change as the coupling parameter δ changes. The
horizontal axis is the coupling parameter δ and the verticalaxis is the average lengths of τ . From this figure, we can seethat the sojourn time between the transitions becomes shorteras increasing the coupling parameter δ.
III. C UBIC MAP
In this study, we consider two coupled cubic maps. Figure 5
shows the cubic map.
The cubic map is described by the following equation:x(n+1) = ax3n − (a + 1)xn(4)where n is an iteration and a is a parameter which determines
the chaotic feature. We can easily confirm that the cubic map
generates various periodic solutions and chaos. Figure 5(a)shows asymmetric 3 periodic solution and Fig. 5(b) shows
symmetric 6 periodic solution.

00.0008

0.001

0.00120.00140.00160.00180.002

deltaFig. 4.
Sojourn time between state transitions versus couplingparameter δ. α = 2.0, β = 4.0, and γ = 0.1.

The followings are the equations of the coupled cubic maps
where  is a coupling parameter.
3 g(x(n) ) = axn − (a + 1)xnx1(n+1) = (1 − )g(x1(n) ) + (g(x2(n) ))(5)
x2(n+1) = (1 − )g(x2(n) ) + (g(x1(n) ))Figure 6 shows examples of the computer simulated resultsof Eq. (5). Figure 6(a) shows the timewaveform obtained withthe parameters giving the periodic solution in Fig. 5(a) and anegative coupling strength. In the same way, Fig. 6(b) shows
the timewaveform obtained with the parameters giving theperiodic solution in Fig. 5(b) and a positive coupling strength.
From the coupled cubic maps, we can observe interesting statesynchronization phenomena. We could confirm that the twomaps synchronize for the parameters giving periodic solutions.

- 92 -xn+1

Secondly, we compare the sojourn time between the statetransitions for the cross-coupled chaotic circuits and the coupled chaotic maps. For the cross-coupled chaotic circuits,
we confirmed that the sojourn time between the transitions
becomes shorter as increasing the coupling parameter asshown in Fig. 4. In contrast, for the coupled chaotic maps,
we confirmed that the sojourn time between the transitions
becomes longer as increasing the coupling parameter as shownin Fig. 7.

xnV. C ONCLUSIONSIn this study, we have investigated the synchronization phenomena of two coupled cubic maps. Also, we have compared
the synchronization behaviors of the cross-coupled chaotic
circuits and the chaotic maps. We could observe interesting
synchronization phenomenon. We confirmed that the two maps
synchronized for the parameters giving periodic solutions. For
the cross-coupled chaotic circuits, the sojourn time between
the transitions became shorter as increasing the coupling parameter. In contrast, for the coupled chaotic maps, the sojourntime between the transitions becomes longer as increasing thecoupling parameter. Clarifying the mechanism of the observed
phenomena is our future work.

(a)
xn+1

xn

R EFERENCES(b)Fig. 5.

Cubic map. (a) a = −3.451. (b) a = −3.70.

However, we do not understand the relationship between theobtainable synchronization states and symmetry property ofthe original solutions.
Figure 7 shows how the sojourn times between the statetransitions change as the coupling parameter  changes. The
parameter a in Fig. 7(a) is the same value as Fig. 5(a). While,
the parameter a in Fig. 7(b) is the same as Fig. 5(b). The
horizontal axis is the coupling parameter  and the verticalaxis is the average lengths of the iterations n between thetransitions. From these figures, we can see that the sojourntime between the transitions becomes longer as increasing thecoupling parameter .
IV. C OMPARISON OF SYNCHRONIZATION BEHAVIORS
In this section, we compare the synchronization behaviors of
the cross-coupled chaotic circuits and coupled chaotic maps.
First, for the cross-coupled chaotic circuits, we can observe
several synchronization phenomena by changing initial conditions. On the other hand, for the coupled chaotic maps,
we can observe anti-phase synchronization for the parameters
giving asymmetric periodic solutions and a negative couplingstrength. Further, we can observe in-phase synchronization
for the parameters giving symmetric periodic solutions anda positive coupling strength. However, we do not understandthe relationship between the obtainable synchronization statesand symmetry property of the original solutions.

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Chaotic Oscillations,” Int. J. Bifurcation and Chaos, vol. 7, no. 3,
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Phenomena in Two Chaotic Circuits Coupled by an Inductor,”
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no. 11, pp. 3423-3433, 2005.
[6] Y. Uchitani, R. Imabayashi and Y. Nishio, “State Transition Phenomenon in Cross-Coupled Chaotic Circuits,” Proc. ofNOLTA’07, pp. 397-400, Sep. 2007.
[7] Y. Uchitani and Y. Nishio, “Investigation of State Transition Phenomena in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’08,pp. 2394-2397, May. 2008.
[8] Y. Uchitani and Y. Nishio, “Synchronization of Small Oscillations in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’09,
pp. 2629-2632, May. 2009.
[9] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations
in a Modified van der Pol Oscillator Containing a Positive
Nonlinear Conductance,” Proc. IEEE, vol. 69, pp. 394-395, 1981.
[10] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuitwith a Negative Resistance and an Ideal Switch of Diodes,” Trans.
of IEICE, vol. E70, no. 8, pp. 744–754, 1987.

- 93 -1.0

x10

-1.0
1.0x2 0-1.0n20000

n20000

(a)
1.0

x10

-1.0
1.0

x2 0-1.0(b)Timewaveform of coupled cubic maps. (a) α = −3.451, and  = −0.0477. (b) α = −3.70, and  = 0.0508.

5000

5000

4500

4500

4000

4000

3500

3500

3000

3000

iteration

iteration

Fig. 6.

2500

2500

2000

2000

1500

1500

1000

1000

500

500

0-0.0484-0.0483-0.0482-0.0481-0.048-0.0479-0.0478-0.0477-0.0476-0.0475-0.047400.05

0.05010.05020.0503(a)Fig. 7.

0.05040.05050.0506

epsilonepsilon(b)

Sojourn time between state transitions versus coupling parameter . (a) a = −3.451. (b) a = −3.70.

- 94 -

0.05070.0508