Spatio-Temporal Phenomena of Coupled Multi-State Chaotic Circuits
Masahiro Wada† and Yoshifumi Nishio‡†Dept. of Intelligence and Informatics, Konan University
8-9-1 Okamoto, Kobe, 658-8501, Japan‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: wada–m@konan–u.ac.jp, nishio@ee.tokushima–u.ac.jp
Abstract
An oscillator including a chaotic system is an important device and one of essential component in the natural
world to solve a mechanism of a nonlinear dynamics ofnetwork. In this study, a simple chaotic circuit with threestates of both chaotic oscillation and limit cycles which isnamed multi-state chaotic circuit is proposed. Synchronization phenomena and complex behavior on a network
system of the multi-state chaotic circuits coupled by inductors are investigated. Several interesting chaotic phenomena of spatio-temporal behavior have been observed in the
coupled network system.
1. Introduction
In recent years, dynamical network system such as
small-world, scale-free and other types is attracted many
researchers’ attention from the view point of complex systems and several engineering applications. It is also important for solving the process of mechanisms in social andbiological systems. We consider that it is a good model
for realization of the real systems to make a coupled network with nonlinear oscillators. Therefore, an oscillator
including a chaotic system is an important device and oneof essential component in the natural world. Nonlinear
dynamics on coupled oscillators is considerable interesting for a wide variety of systems in several scientific fields
and some engineering applications, especially in stochastic models. Thought, many types of coupled circuit systems have been widely studied in order to clarify inherent features and many researchers have already proposed
and investigated mechanism of them. The dynamics ofspatio-temporal phenomena or phase synchronization on
several coupled systems is still considerable interest fromthe viewpoint of both natural scientific fields and severalapplications. They have been confirmed in several coupled chaotic systems [1]–[3]. Phase synchronization andpattern dynamics are also interesting for several engineering applications. On the other hand, many types of chaoticsystems and circuits have already been proposed and investigated in detail. As interesting phenomena, there are
famous chaotic attractors such a double-scroll family [4],n-double scroll [5]–[7] and scroll grid attractors [8]. If

the active elements including in the systems have complexity constructed by compound some nonlinear elements, it
can be easily considered that they yield several interesting features. In our previous studies, the circuit which
can individually behave both chaotic or periodic oscillations in the same parameters had been investigated [9]-[13].
This type of circuit was called a multi-state chaotic circuit(abbr. MSCC). Multimode oscillations in coupled two or
more multi-state chaotic circuits had been shown [10] on
real circuit. A complicated and interesting phenomena ofphase synchronization had also been investigated [11]-[13].
It is known that complex behavior can be confirmed suchchaotic itinerancy and spatio-temporal chaos on the large
scale coupled networks.
In this study, several synchronization phenomena andcomplex behavior on coupled network system of multistate chaotic circuits are investigated. This paper shows
several phase synchronization phenomena of multi-statechaotic oscillators coupled by some inductors as a simple network. Each oscillator circuit can individually behave both limit cycles and chaos in the same parameters.
This proposed circuit can behave three steady states of bothchaotic and two different size of limit cycles when an appropriate initial conditions are supplied. We consider acoupled chaotic network system which each chaotic oscillator is connected to neighbors by inductors. In numericalsimulation, many types of phase synchronization modes are
asynchronously confirmed in the proposed systems, further
all parameters are the same. This means several phase synchronization modes are coexisting in the same parameters.
Several interesting chaotic phenomena of spatio-temporal
behavior have been observed in the coupled network system.
2. Model description of an MSCC
An MSCC is a basic circuit as a subcomponent of acoupled network model. The designed circuit is shown inFig. 1. In this study, we substitute a symmetrical continuous piecewise linear resistor for the negative active resistor
including in the original chaotic circuit. The piecewise linear resistor can be easily constructed by combining somecomponents in parallel [9][10]. By changing the following
variables and parameters as follows- 23 -

vdi L1

VdL2+vi L2NRCCL1C-rd / 2 = r si

D-Vd

Figure 1: A multi-state chaotic circuit with piecewise linear resistors NR and v − i characteristic of the diode D.

Figure 3: Attractor drawing onto the z − x plane for theparameters β = 10.0, γ∗ = 0.78 and δ = 100. h(z): {p1 ,p2 , p3 , p4 }={0.65, 0.55, 0.40, 0.30}, {m0 , m1 , m2 , m3 , m4 }= {-1.0, 2.0, -1.0, 1.0, -0.15}.
Figure 2: Design for sawtooth nonlinear resistor NR in thecircuit.
rr
CC
Vd x , iL2 =Vd y ,L1
L1pd,v = V d z , t = L1 C τ , “ · ” =(1)rrdτ
L1L1Cβ=, γ=g, δ = rs
L2
C
L1
where g is a linear negative conductance value of NR if weconsider the negative resistor as an ideal element. Consider that the part of negative resistance NR in Fig. 1 replaces to the function h(z) represented by a voltage sourcez as canonical form with 9-segments as shown in Fig. 2.
When we chose the threshold voltage Vd for a normalized
parameter, then the circuit equations can be normalized andrewritten as follows.
ẋ = z





(2)
ẏ = β z − f (y) ż = −(x + y) − h(z)1
f (y) = |δy + 1| − |δy − 1|
(3)
2
 K


γ∗ 
X
∗
(m−m)
p|−|z+p|}h(z) = m0 γ z+ {|z−
kk+1
k+1k+12 
k=0(4)
where f (y) is a function of the current y and h(z) is a function of the voltage z, respectively. The function h(z) which
is designed for several segment piecewise linear as symmetric with respect to the origin. The parameter γ∗ is used
for a basic common value, hence the values mk (k = 0, 1, 2,· · ·, K) mean magnitude of the slope to the ratio for γ∗ .
iL1 =Figure 4: A coupled network model which each circuit isconnected to four neighbors.
Figure 3 also shows a typical chaotic attractor obtained
for the parameters β = 10.0, γ∗ = 0.78, δ = 100, withpiecewise linear characteristics realized by breakpoints p1= 0.65, p2 = 0.55, p3 = 0.40, p4 = 0.30, slopes m0 = −1.0,
m1 = 2.0, m2 = −1.0, m3 = 1.0 and m4 = −0.15. We canconfirm that chaotic and two periodic attractors coexist in
the circuit. This means coexistence of both chaos and twodifferent size of limit cycles in the same parameters.
3. Simulation for coupled MSCCs on a network model
In this section, the model of coupled MSCCs by inductors are investigated. For example of a coupled network,
now let us consider the coupled MSCCs model which combined number of N × M chaotic circuits are connected byinductors L0 to neighbors’ circuit as a network structure
shown in Fig. 4. The circuit index is defined as (i, j). It- 24 -

is note that every chaotic circuit is composed by all thesame parameters and connected to four neighbors circuits.
The circuit on the edge of this coupled network is connected to an opposite side circuit, it seems like a distribution on surface of the torus structure. By the similar way
described before, the circuit equations of coupled MSCCs
can be normalized by the variables (1) with a new parameter α = L1 /L0 . Therefore, the whole circuit equations canbe rewritten as follows.
ẋ(i, j)





ẏ(i, j)







ż(i, j)








===z(i, j)
β z(i, j) − f (y(i, j) )

X


α xΦ − 4x(i, j) ΦAcknowledgmentThis work was partly supported by GRANT-IN-AID for”Open Research Center” project from MEXT, Japan.
References[1] Y. Nishio, A. Ushida, “On a Ring of Chaotic Circuits
Coupled by Inductors,” IEICE Trans., vol. E78-A, no.
5, pp. 608–617, 1995.
[2] Y. Nishio, A. Ushida, “Phase Synchronization in a
Ring of Chaotic Circuits,” Proc. of NOLTA’01, pp.
163–166, 2001.

(5)−(x(i, j) + y(i, j) ) − h(z(i, j) )where functions f (y) and h(z) are similar to (3) and (4),respectively. Φ means a set of four neighbors to x(i, j) .
We show some computer calculation results by using
4-th order Runge–Kutta method with time step size ∆t =0.001 for the circuit equations in some cases of (N, M) asfollows. The parameters of each circuit are the same in theSec. 2. The initial conditions of each circuit are supplied
at random. If the case (N, M) = (2, 2), this coupled system is corresponding to the case of a ring structure in [12].
We show some computer simulation results for some cases.
Figure 5 shows some typical results obtained from computer simulation in some cases of coupled number (N, M).
The left part of the figure shows attractors of the circuit,and the right part indicates synchronization state between
two circuits. We can confirm complex and chaotic synchronization phenomena on the coupled system. Further
we can several types of complex and interesting synchronization phenomena in the same parameters, i.e., in-phasesynchronization, anti-phase synchronization, clustering ofphase synchronization, phase locking and other types. Several phase synchronization modes are coexisting in spite ofthe same parameters.

[3] Y. Nishio, A. Ushida, “Statistical Analysis of Chaotic
Wandering in Simple Chaotic Circuits,” Proc. of ITCCSCC’01, pp. 1034–1037, 2001.
[4] L.O. Chua, M. Komuro, T. Matsumoto, “The DoubleScroll Family,” IEEE Trans. Circuits Syst.–I, vol. 33,
no. 11, pp. 1072–1118, 1986.
[5] J.A.K. Suykens, J. Vandewalle, “Generation of nDouble Scrolls (n = 1,2,3,4,...),” IEEE Trans. Circuits
Syst.–I, vol. 40, no. 11, pp. 861–867, 1993.
[6] M. Wada, Y. Nishio, A. Ushida, ”Chaotic Itinerancy
Phenomena on Coupled n-Double Scrolls ChaoticCircuits,” Proc. of ISCAS’99, vol. 5, pp. 487–490,1999.
[7] G. Zhong, K. Man, G. Chen, “A Systematic Approachto Generating n-Scroll Attractors,” Int. J. of Bif. andChaos, vol. 12, no. 12, pp. 2907–2915, 2002.
[8] M.E. Yalçin, J.A. Suykens, J. Vandewalle, “Familiesof Scroll Grid Attractors,” Int. J. of Bif. and Chaos,vol. 12, no. 1, pp. 23–41, 2002.
[9] M. Wada, Y. Nishio, “Coexistence Phenomena ofChaotic and Non-Chaotic Oscillations in a ThreeDimensional Chaotic Circuits,” Proc. NOLTA2002,
vol. 2, pp. 869–872, 2002.
[10] M. Wada, Y. Nishio, “Multimode Asynchronous Oscillations in Coupled Multi-State Chaotic Circuits,”
Proc. of NCSP’04, pp. 535–538, 2004.

4. Conclusions
In this study, we have investigated several synchronization modes in coupled multi-state chaotic oscillator circuits. Coexistence of several types oscillation modes havebeen confirmed in coupled MSCCs by inductors as a coupled network systems. Several interesting chaotic phenomena of spatio-temporal behavior have been observed in
the coupled network system. On a large scale of coupledchaotic oscillators such a small-world and a scale-free network, we consider that several types of complex behavior
are expected to yield novel applications and inherent emergent properties in the natural systems.

[11] M. Wada, Y. Nishio, “Several Phase Synchronization Modes in Coupled Multi-State Chaotic Circuits,”
Proc. of NOLTA’04, pp. 525–528, 2004.
[12] M. Wada, Y. Nishio, “Coexistence of Several Phase
Synchronization in a Ring of Coupled Multi-State
Chaotic Oscillators,” Proc. of NOLTA’07, pp. 393–396, 2007.
[13] M. Wada, Y. Nishio, “Phase Synchronization Phenomena on Coupled Multi-State Oscillators,” Proc. ofNCSP’08, pp. 144–147, 2008.

- 25 -(d)
(a)

(e)

(b)

(c)

(f)

(g)
Figure 5: Simulation results obtained from coupled MSCCs for α = 0.10, β = 10.0, γ∗ = 0.78, δ = 100. h(z): {p1 , p2 , p3 ,p4 }={0.65, 0.55, 0.40, 0.30}, {m0 , m1 , m2 , m3 , m4 } = {-1.0, 2.0, -1.0, 1.0, -0.15}. Size (N, M) of the network: (a) (2,2), (b)(2,3), (c) (2,4), (d) (3,3), (e) (3,4), (f) (4,4), and (g) (4,5) with waveform of each oscillator.

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