Proceedings of NOMA'13 Workshop
September 3-4, 2013, Zaragoza, Spain

Synchronization of Switching Phenomena
Observed in Discrete and Continuous
Coupled Chaotic Systems
Takuya Nishimoto† , Yasuteru Hosokawa‡ , Yoko Uwate† and Yoshifumi Nishio††Tokushima University
Minami-Josanjima, Tokushima, 770-8506
Email: {nishimoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— Synchronization of switching phenomena can beobserved in a few coupled chaotic systems. Synchronizationof self-switching phenomena have reported by Sekiya et al.
Synchronization of self-switching phenomena is generated whenchaotic elements are asynchronized. In our past study, we
have observed similar phenomena on coupled chaotic circuitswith Nishio circuit. Although coupling topology and coupled
elements are different each other, synchronization of switching
phenomena have observed in each system. In this study, we
have investigate the phenomena on coupled chaotic maps. This
study shows that the phenomena can be observed in discrete andcontinuous system. Synchronization of self-switching phenomena
have generality and may be observed in many natural system.

I. I NTRODUCTION
There are many of nonlinear phenomena in the naturalworld. For example, chaos synchronization, clustering phenomenon, and so on. It is important that grasping the mechanism of those phenomena to clarify nonlinear phenomena exist
in the natural world. Especially, synchronization phenomena
observed in coupled chaotic systems have been attracted
attentions since these system might be regarded as the models
of natural system and give contribution to clarify the highorder complex nonlinear phenomena.
On the other hand, some chaotic elements have coexisting
attractors. Double-scroll attractor can be observed by changing the parameters for those chaotic elements. In chaoticoscillators, double-scroll means the behavior that the solutionswitches two attractors without external force. This phenomenon is called “self-switching phenomenon”. Moreover,”synchronization of self-switching phenomena” have reported
by Sekiya et al[1]. The phenomena mean that the switchingof attractor is occur synchronously in each subcircuit though
the system keeps asynchronous state. It is very interesting that
there are simultaneity though the system behaves disorderly.
We have observed similar phenomena in simple coupledchaotic circuit[2]. The study shows that the phenomena havegenerality and might be observed high-order coupled chaoticsystem. It should be noted that the coupled chaotic system,
the chaotic element of which are included in each study, hasdouble-scroll attractor. In this study, we investigate synchronization of switching phenomena on coupled chaotic map. The‡Shikoku University
Furukawa, Ohjin, Tokushima, 771-1192
Email: hosokawa@keiei.shikoku-u.ac.jp

investigated system is composed with a cubic function. Thischaotic element have symmetrical structure and the element
which is used in previous study also have same characteristic.
Namely, it is considered that a cubic function belongs todouble-scroll family. Although coupled chaotic map is discrete
system, synchronization of self-switching phenomena haveobserved. This study indicates that the phenomena have broad
generality, and gives contribution to clarify the mechanism ofsynchronization of swithcing phenomena.
II. C OUPLED C HAOTIC S YSTEMS
Coupled chaotic system shown in Fig. 1 is investigated inthis study. Chaotic elements are shown in Fig. 2 (a) and (b).
Figure 2 (a) is a simple chaotic oscillators proposed by Nishio
et al. Equations for each system are described as follows. InFig. 1.

System model.

chaotic element as shown Fig. 2, by changing parameters and49

i n1i n2C

L n2vnL n1-rf( i n2 )(a)Fig. 2.

(b)

Chaotic elements. (a) Chaotic circuit. (b) Cubic function.

Proceedings of NOMA'13 Workshop
September 3-4, 2013, Zaragoza, Spain

TABLE I

variables as follows,√t = L1 Cτ, in1 = VthATTRACTOR - STATE DEFINITION .

√Cxn , in2 = yn ,L1 √dC′vn = Vth zn , ”· ” =, α=g, vdn = Vth vdndτ
L1√√√
1L1 CCL1, γ=and δ = Gβ = VthL2
Vth L1C

Attractor-state{Red, Red}
{Red, Blue}(4)

Figure 4 shows simulation results of each system model.
These figures show time waveforms Fig. 4(1) and lissajous
figure and scatter diagram Fig. 4(2). In the time waveforms,
there red and blue waves that are show attractor state. Namely,
the change of color means switching phenomena have occurred. Green line also shows a difference of attractor state.
Synchronization of switching phenomena occurred in Fig. 4.
In each simulation, the amplitude of z1 − z2 or xn (1) − xn (2)is observed that denote each system keeps asynchronous state.
Although, asynchronous state is kept at each system, switching phenomena are synchronized. We call such phenomena”synchronization of switching phenomena”. These phenomena
are collapsing according to decrease the coupling strength. In
circuit experiment, similar phenomena also can be observed
as shown Fig. 5. It should be noted that anti-phase switching
is observed in both systems.

(5)

(A)

k=1

We approximate i-v characteristic of bi-directionally coupled
diodes as following function.
′vdn= (2u(yn2 ) − 1)

(3)

Chaotic element as shown Fig.2 (b) is written as
f (x) = −ax + ax3 .

IV. I N CASE OF N = 2.

(1)

the system equation of Fig. 2 (a) is described as follows:ẋn = αxn + zn′ ẏn = β {zn − vdn}
(2)
()
N
∑
zk
 żn = −xn − γyn − δ N zn −StateA
BThen, system equation is described as follows:ε ∑f (xn (j)).
xn+1 (i) = (1 − ϵ)f (xn (i)) +N j=1
Nz1
Z1z2
Z2z2
Z1z-1 Z2

Where N and u() denote the number of chaotic elements andstep function respectively.
III. C OEXISTING ATTRACTORS
Figure 3 shows the bifurcation diagram and coexisting
attractors of chaotic element as shown Fig. 2 (a) and (b)respectively. Each bifurcation diagram is described with different initial values. Period-doubling bifurcation have begun atdifferent point of the vertical because these chaotic element are
the double-scroll family. Coexisting attractors of each element
are shown in Fig. 3 (2). We distinguished each coexisting
attractors from bifurcation diagram as follows.
In chaotic element (a),(1) The Poincaré section is defined as zn = 1.0 andẏn < 0.
(2) When the solution hits the Poincaré section, andxn ≤ 0.675, the color is set as red. In the case ofyn > 0.675, the color is set as blue.
In chaotic element (b),(1) When xn (i) ≤ 0, the color is set as red. In the caseof xn (i) > 0, the color is set as blue.
These definitions are important to investigate switching of
attractors and as such, are being applied to the simulationsof this study.

(B)

Xn(1)
Xn(2)

Xn(1) - Xn(2)

(1)

(2)

Fig. 4. Computer simulation result. (A) Chaotic element as shown Fig. 2(a).
N = 2, α = 0.40, β = 3.0 and δ = 0.2. (1) Time waveforms of zi
and z1 − z2 . (2) z1 vs z2 . (B) Chaotic element (b). N = 2, a = 2.8 andε = 0.875.(1) Time waveforms of xn (i). (2) Scatter diagram of xn (1) vsxn (2).

V. M ECHANISM OF S YNCHRONIZATION OF
S WITCHING P HENOMENA
In this section, we investigate the mechanism of synchronization of switching phenomena. Figure 6 shows thedistribution of xn (1). Note that the requirement of switchingis when the element satisfy |xn (i)| > 1. Blue pins show thedistribution of xn (1) in case of state A, red pins show incase of state B. Table. I shows state definition of attractor.
Red pins are sifted to center of the distribution compare withblue pins. Namely, in case of state B, switching phenomena

50

Proceedings of NOMA'13 Workshop
September 3-4, 2013, Zaragoza, SpainZ
Z
1.5

1.5

(A)

-1.5

YY

-1.5

(B)

(1)

(2)

Fig. 3. Computer simulation results. (A) Chaotic element as shown in Fig. 2(a). (1)Bifurcation diagram. β = 3.0. Horizontal: α. (2) Coexisting attractors.
α = 0.4 and β = 3.0. (B) Chaotic element as shown Fig. 2(b).(1) Bifurcation diagram. Horizontal: a. (2) Coexisting attractors. a = 2.8.

Fig. 5.
Circuit experiment of coupled chaotic circuit of Fig. 2(a). Vertical:vn [2V/dvi], Horizontal:time[10ms/dvi]. N = 2, L1 =642.8[mH], L2 = 211.1[mH], C = 7.06[nF], G = 0.089[mS].

hardly occur than state A. Figure 8 shows the probabilityof switching. We measured the probability of switching in
1 < n < 100000. Both state are decreased according toincreasing the coupling strength. However, state A have acertain probability of switching independent of ϵ. On thecontrary, the probability is decreased substantially according
to increasing the coupling strength. Namely, state B is stablecompare with state A for this system. Such behavior can be
explained as follows. In case of state A, each element has
positive value. Therefore probability of switching is increased
since the absolute value of xn (1) become large. In case ofstate B, an element takes positive value while the other takes
negative value. For such occasion, the absolute value of xn (1)become small and probability of switching is decreased. It isconsidered that the mechanism of switching phenomena inthis system. Figure 7 shows the probability of switching per1τ in the system of Fig. 2 (b). We measured the probability
in 2000 < τ < 2000000. The probability is also decreased
by increasing the coupling strength. We assume that there aresimilar mechanisms in the system.

51

Proceedings of NOMA'13 Workshop
September 3-4, 2013, Zaragoza, Spainz
Z2z 2Z3z 3Z1 1z zz3
Z1z-1 Z3z3
Z2z-2 Z3

Z1 -1Z2 2Fig. 6.

Distribution of the value. a = 2.8, ε = 0.875.
Fig. 9. Computer simulation resul of chaotic element as shown Fig. 2(a).
N = 3, α = 0.4, β = 3.0 and δ = 0.17.

Xn(1)
Xn(2)
Xn(3)
Xn(1) - Xn(2)
Xn(1) - Xn(3)
Xn(2) - Xn(3)Fig. 10. Computer simulation resul of chaotic element as shown Fig. 2(b).
N = 3, a = 2.8, δ = 0.105.
Fig. 7. Probability of switching of element as shown Fig. 2(a). α = 0.4,β = 3.0.

switching.
VII. C ONCLUSIONIn this study, synchronization of switching phenomena haveinvestigated in continuous and discrete systems. As a result,anti-phase synchronization of switching phenomena have observed in each system. Additionally, mechanism of the phenomena have declared in discrete system. This mechanism givecontribution to clarify the phenomena observed in continuoussystem. These results denote that synchronization of switching
phenomena may be observed in other natural system generally.
R EFERENCES
Fig. 8.

Probability of switching of Fig. 2(b). a = 2.8.

VI. I N CASE OF N = 3
Figure 10 shows simulation results of coupled chaotic map
in case of N = 3. Synchronization of switching phenomena
can be observed, however, the phenomena is not occurred constantly. Additionally, the simultaneity of switching is slightly
collapsed. Figure 9 shows simulation result of coupled chaoticcircuit. In this case, the phenomena are not also occurredconstantly. Anti-phase switching is observed in both systems.
In case of N = 3, the systems obey the mechanism explained insection V . In the system proposed by Sekiya et al, synchronization of switching phenomena is observed with in-phase[1] H. Sekiya, S. Mori and I. Sasase, “Synchronization of Self-Switching
Phenomena on Full-Coupled Chaotic Oscillators,” IEICE Trans., vol. J83A, no. 11, pp. 1264–1275, 2000.
[2] T. Nishimoto, Y. Hosokawa and Y. Nishio, “Anti-phase Synchronizationof Switching Phenomena in Globally Coupled System of Chaotic Circuits,” IEICE Technical Report, no. NLP-48 & CAS2010-32, pp. 1–4,2010.
[3] T. Nishimoto, Y. Hosokawa and Y. Nishio, “In-Phase and Anti-Phase
Synchronization of Switching Phenomena in Coupled Chaotic Circuits”
Proc. of IEEE Workshop on Nonlinear Circuit Networks (NCN’10) ,pp. 15–18, 2010.
[4] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses of Windows
in a Symmetric Circuit,” IEEE Trans. Circuit and Systems, vol. 37, no. 4,
pp. 473–487, 1990.

52