2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012

Coexistence Phenomenon Observed in Coupled Chua’s Circuits Involving Parameter Varying
Hironori Kumeno†‡ , Danièle Fournier-Prunaret‡ and Yoshifumi Nishio††Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {kumeno, nishio}@ee.tokushima-u.ac.jp

Abstract
In this study, we investigate behaviors of two identical coupled Chua’s circuits whose parameters periodically varying inassociated with each of the period of internal state values. For
the periodically varying of the parameters, odd-order cycles
do not exist in this system. We observed coexistence of lots of
attractors which have different orders of cycles and different
synchronization states.
1. Introduction‡LAAS-CNRS
7 avenue du colonel Roche,
F-31077 Toulouse Cedex 4, France
Email: daniele.fournier@insa-toulouse.fr

nal state value and the two circuits are coupled by a resistor
in parallel. In the next section, we propose the coupled parametrically forced Chua’s circuit. In section 3, we investigate
bifurcations in the non-coupled parametrically forced Chua’scircuit. Non-existence of odd order cycles and coexistenceof different attractors are observed. In section 4, we consider
the coupled parametrically forced Chua’s circuits. Coexistingof many attractors whose synchronizations states are different
are observed. The last section is devoted to the conclusion.
2. Parametrically Forced Chua’s Circuit

Chaotic behaviors are observed in various research fieldand cause interesting and complex phenomena. Coupledchaotic systems attract extensive attention as good models
which describe the complimented phenomena in the naturalworld. The field of coupled chaotic systems has been developed since the discovering of synchronization of chaotic trajectories [1]. Studies about the coupled systems are carriedout in various field, such as in physics [2], biology [3, 4] andengineering [5]. Furthermore, researchers suggest that synchronization phenomena of coupled systems have some relationships with information processing in brain. So, to investigate the coupled chaotic system is important to understandingof information processing mechanism of the brain.
By the way, it has been reported that if an autonomic
neuron is influenced by external force, its characteristics arechanged such as the neuron behaves periodic or chaotic [6].
How to give periodic external force is thought about in various ways, but we focus on a system whose parameter is forced
into periodic varying and investigate behavior of the system
and its coupling. The parametric force almost causes periodic oscillation and chaos in a simple oscillator [7, 8, 9]. Previously, we have investigated coupled logistic maps whoseparameters are forced into periodic varying and observed interesting characteristic behaviors of the parametrically forced
system [10].
In this study, we investigate behaviors of a two-coupledparametrically forced Chua’s circuit. In the system, the linear
resistor of each parametrically forced Chua’s circuit is forced
into periodic varying in associated with the period of its inter-In this study, we consider a coupled continuous-time system whose parameter is forced into periodic varying. Chua’s
circuit is used as the continuous-time system. Tho Chua’s circuits are coupled by a resistor in parallel. The circuit modelof the coupled system is shown in Fig. 1. For this coupling,v1k (k = 1, 2) affect to the other subcircuits with the coupling
intensity corresponding to the coupling resistor. The linear
resistors of the Chua’s circuits alternately changes from R1
to R2 depending on v2k . The switches which cause periodically varying of the resistors are controlled by v2k . Namely,
the switch of each of the subcircuits is controlled by the statevalue of each subcircuit. The switches shift when v2k is equal
to 0 and changes from negative value to positive value. Figure 2 shows the relationship between v21 and the motion ofthe switch. The linear resistor are changed every one periodof v2k .
The state equations of the parametrically forced Chua’s circuit are:{}1
11dv1k=(v2k − v1k ) − f (v1k ) + (v1l − v1k )


dt
C1 Rr



{}


1
 dv2k = 1(v1k − v2k ) + i3k
dt
C2 R,


1di3k



= − v2k

dt
L


(k = 1, 2)(l = 1, 2, l 6= k)(1)- 603 -

(a)Figure 2: The motion of the switch depending on v2 .

We investigate bifurcations in the non-coupled parametrically forced Chua’s circuit by carry out computer simulations
with fourth-order Runge-Kutta method when parameters are
set as α = 3.7, β = 4.5, Ga = −1.3 and Gb = −0.6. Figure 3 shows one-parameter bifurcation diagrams when R1 isfixed and R2 varies. In Fig. 3(a), an order two cycle started(b)from the right side of the figure continues to undergo flip biFigure 1: The circuit model of the parametrically forced furcations and the bifurcation cascade give rise to chaos. On
Chua’s circuit and the i − v characteristics of the nonlinear the cascade, an order 1 periodic cycle appears at the point
R1 = R2 . Except that point, all cycles are even order. Inresistor NR .
Fig. 3(b), the order of an cycle started from R1 = R2 increases by both of increase and decrease of R2 when R2
crossing through flip bifurcations. In the basic Chua’s circuit,
where, R alternately changes from R1 to R2 depending on by decreasing the resistance of the linear resistor, the orders
v2k ; andof cycles simply increase when the cycles undergo flip bifurcations. Whereas, in the parametrically forced Chua’s circuit,
1f (v1k ) = Gb v1k + (Ga − Gb ){|v1k+ E| − |v1k − E|} (2) by decreasing R2 , the orders of cycles increase and decrease2
when the cycles undergo flip bifurcations.
is the v − i characteristic of the nonlinear resistor NR with
Figure 4 shows attractors and their Poincaré maps on aa slope equal to Ga in the inner region and Gb in the outer cascade when R is fixed and at 1.10 and R decreases. In
12region. A typical v − i characteristic of NR is shown in Fig. 4(a) and (a’), the parameters are the same, whereas initialFig. 1(b). By using following parameters and variables:
values are different. Two order 2 cycles coexist. By decreas
ing R2 , the cycles undergo a flip bifurcation and then become
1C2C2


t, α =,β = τ=order 4 cycles as shown in Fig. 4(b) and (b’). The cycles conRC2
C1L(3) tinue to undergo flip bifurcations and then give rise to chaos.
 xk = v1k , y = v2k , z = R i3k ,For some parameter regions, different order cycles coexist,
EE
Efor instance, an order 6 cycle and a chaos coexist as shownin Fig. 4(c) and (c’). More decreasing R2 , one of the coexthe normalized circuit equations are given as:
isting attractors disappears and a chaotic attractor remains as

{}
Rdx

kshown in Fig. 4(d). Finally, a double-scroll chaotic attractor= α (yk − xk ) − Rf (yk ) + (xl − xk )dτrisobserved as shown in Fig. 4(e).
dyk.
= xk − yk + zk
dτ
4. Synchronization in the Coupled Systemdzk2= −βR ykIn this section, we investigate coexistence of attractors obdτ(4) served in two coupled parametrically forced Chua’s circuits
when the parameters are set as α = 3.7, β = 4.5, Ga =−1.3, Gb = −0.6, R1 = 1.07, R2 = 1.0676 and r = 4.0.
3. Bifurcation Analysis in the Non-Coupled SystemFor the parameters of the subcircuits, order 2 cycle and order- 604 -

(a)

(a’)

(b)

(b’)

(c)

(c’)

(d)

(e)

(a)

Figure 4: Attractors observed in the computer simulations for
α = 3.7, β = 4.5, Ga = −1.3, Gb = −0.6 and R1 = 1.07.
(a)R2 = 1.08. (b)R2 = 1.07. (c)R2 = 1.056. (d)R2 = 1.05.
(e)R2 = 1.04.

(b)

low and green colored regions correspond that the trajectories
started from those regions converge the attractors shown in
Figure 3: One-parameter bifurcation diagrams for the fixed Fig. 5(1), (2), (3) and (4), respectively. Geometric patterns
parameters α = 3.7, β = 4.5, Ga = −1.3 and Gb = −0.6 are presented fractal structures are confirmed.
and variant R2 . (a) R1 = 1.10. (b) R1 = 1.07
5. Conclusions
4 cycle coexist in the non-coupled case.
Figure 5 shows attractors and phase differences observedin computer simulations. Four attractors coexist as shown inFig. 5. Three of the four coexisting attractors are order 2 cycles, and the remain coexisting attractor is order 4 cycle. In
any case of the coexisting attractors, the two subcircuits aresynchronized at the in-phase. Here, we focus on the motionsof the switches. We can see that there are two pattern of themotion of the switch that the two switches shift at the in-phaseand the opposite-phase. By the coupling, the motions of theswitches are attracted at the in-phase or the opposite-phase.
For the motions of the switches, lots of stable states are constructed.
Figure 6 shows basins on (x1 , z1 ) phase plane when y1 = 0
and (x2 , y2 , z2 ) = (0.3828, 0, 0.485) which is a coordinateof the orbit shown in Fig. 5(2). In the figure, red, blue, yel-In this study, we investigated synchronization in two identical coupled Chua’s circuits whose parameters periodicallyvarying in associated with each of the period of internal statevalues. For the periodically varying of the parameters, oddorder cycles do not exist in this system. We have confirmed
coexistence of lots of attractors which have different orders ofcycles and different synchronization states. By the coupling,
the motions of the switches are attracted at the in-phase or theopposite-phase. Then, the motions of the switches cause lotsof stable states.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.

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(2)

(3)

(4)

Figure 5: Attractors and phase differences in the coupled parametrically forced Chua’s circuit.

[4] Cosp, J. and Madrenas, J. and Alarcon, E. and Vidal, E.
and Villar, G., “Synchronization of nonlinear electronic
oscillators for neural computation.” IEEE Trans. NeuralNetworks, vol. 15, pp. 1315-1327, 2004.
[5] Tanaka, G. and Aihara, K., “Multistate Associative Memory with Parametrically Coupled Map Networks.” Int. J.
Bifurcation and Chaos, vol. 15, num. 4, pp. 1395-1410,2005.
[6] Hayashi, H. and Ishizuka, S. and Hirakawa, K. “Transition to chaos via intermittency in the Onchidium pacemaker neuron.” Physics letters. A, vol. 98, pp. 474-476,
Nov, 1983.

Figure 6: Basins on (x1 , z1 ) phase plane when y1 = 0 and(x2 , y2 , z2 ) = (0.3828, 0, 0.485)

References[7] Hayashi, C., Nonlinear Oscillations in Physical Systems,
Chap. 11, McGraw-Hill, New York 1964.
[8] Hayashi, C. and Abe, M. and Oshima, K. and Kawakami,H., “The Method of Mapping as Applied to the Solution
for Certain Types of Nonlinear Differential Equations,”
Proc. of the Ninth International Conference on Nonlinear
Oscillations, pp. 1-8, 1981.

[1] Fujisaka, H. and Yamada, T., “Stability theory of syn[9] Inoue, M., “A Method of Analysis for the Bifurcation
chronized motion in coupled-oscillator systems.” Prog.
of the Almost Periodic Oscillation and the Generation ofTheor. Phys., vol. 69, pp. 32-47, 1983.
Chaos in a Parametric Excitation Circuit,” Trans. of IEICE, vol. J68-A, no. 7, pp. 621-626, 1985.
[2] Lind, P. G. and Titz, S. and Kuhlbrodt, T. and Corte-real,J. A. M. and Kurths, J. and Gallas, J. A. C. and Feudel,[10] Kumeno, H. and Nishio, Y. and Fournier-Prunaret, D.,
U., “Coupled Bistable Maps: A Tool to Study Convec“Bifurcation and Basin in Two Coupled Parametricallytion Parameterization in Ocean Models.” Int. J. BifurcaForced Logistic Maps,” Proc. of IEEE International Symtion and Chaos, vol. 14, num. 3, pp. 999-1015, 2004.
posium on Circuits and Systems (ISCAS’11), pp. 13231326, May. 2011.
[3] Freeman, W. J., “A Neurobiological Theory of Meaning
in Perception. Part I: Information and Meaning in Non- [11] Mira, C., Chaotic Dynamics. chap. 6, World Scientificconvergent and Nonlocal Brain Dynamics.” Int. J. BifurPublishing Co. Pet. Ltd., Singapore, 1987.
cation and Chaos, vol. 13, num. 9, pp. 2493-2511, 2003.

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