2009 International Symposium on Nonlinear Theory and its Applications
NOLTA'09, Sapporo, Japan, October 18-21, 2009

Phase Synchronization in a Coupled Non-Uniform Networkof Multi-State Chaotic Oscillators
Masahiro WADA† and Yoshifumi NISHIO‡†Department of Information Science and Systems Engineering, Konan University
8–9–1 Okamoto, Higashinada-ku, Kobe 658-8501, Japan‡Department 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—This paper presents several phase synchronization modes of multi-state chaotic oscillators coupled
as a non-uniform network system. They have been known
several phase patters in several oscillators. Each chaotic
circuit which used in this paper can individually behaveboth chaotic and two periodic oscillations in the same parameters asynchronously. In this study, such the coupledchaotic circuits are proposed and classifications of phasesynchronization modes are investigated. In numerical simulation, many types of phase synchronization modes could
be confirmed.
1. Introduction
Nonlinear dynamics on coupled chaotic oscillators is
considerable interesting for a wide variety of systems inseveral scientific fields and applications. Many types ofcoupled systems have been widely studied in order to clarify inherent features and many researchers have alreadyproposed and investigated them. Coupled chaotic systems
are as one of them which have several varieties of interesting behavior with emergent properties. The dynamics ofchaotic multimode oscillations or chaotic itinerancy on several coupled systems is still considerable interest from theviewpoint of both natural scientific fields and several applications. They have been confirmed in several systems;e.g., coupled van der Pol oscillators [1], coupled chaoticsystems [2], and so on. Phase synchronization and pattern
dynamics are also interesting for several engineering applications. On the other hand, many types of chaotic systems
and circuits have already been proposed and investigated indetail. As interesting phenomena, there are famous chaotic
attractors such a double-scroll family [3], n-double scroll[4]–[6] and scroll grid attractors [7]. If the active elements
including in the systems have complexity constructed bycompound some nonlinear elements, it can be easily considered that they yield several interesting features. The circuit which can individually behave both chaotic or periodic oscillations in the same parameters had been shown[8]. This type of circuit was called a Multi–State Chaotic
Oscillator/Circuit (abbr. MSCO/MSCC). Multimode oscillations in coupled two or more multi–state chaotic circuits
had also been investigated [9]–[11]. Furthermore, we have

vdi L1

VdL2+vi L2NRCCCL1-rd / 2 = r siD-VdFigure 1: Multi-state chaotic oscillator with piecewiselinear resistors NR and v-i characteristic of the diode D
been reported several phase synchronization on a uniform
network system [12]. However they have been treated onlya uniform network of a large scale system. It is very important to investigate a non-uniform system of the coupled
network in the sense of solving the several phenomena of
the natural world.
In this study, several phase patterns and multimode asynchronous oscillations on the coupled MSCOs are investigated. There is a typical three dimensional autonomouschaotic system, which consists of three memory elements,
some diodes and designed negative resistors. We substitutea symmetrical continuous segments piecewise linear resistor for the negative active resistor including in the original chaotic circuit. This proposed circuit can behave bothchaotic and periodic oscillations in the same parameterswhen we supply with different initial conditions. Several
phase synchronization and classification of several phasepatterns in some MSCOs coupled as a non-uniform network are investigated. It will be shown that several typesof phase synchronization modes can be confirmed asynchronously, but all parameter settings of each circuit arethe same.
2. Model Description
The circuit shown in Fig. 1 is modified chaotic circuitfrom a well–known three dimensional chaotic circuit [13].
The original circuit consists of three memory elements,
some diodes and designed negative resistors. It is wellknown that it can behave as Rössler type chaotic motions.
We substitute a symmetrical piecewise linear resistor forthe negative active resistor including in the original chaotic- 222 -

h(z)

0.5

0−0.5
−1−0.500.5

1z

Figure 2: Designed sawtooth nonlinear resistor NR in thecircuit. 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}
circuit. Further this circuit possesses another symmetricalpiecewise nonlinear resistor with respect to the origin.
An MSCO which is modified from the original circuit asshown in Fig. 1 is a basic circuit as a subcomponent of acoupled network model. 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 bycombining some components in parallel [8].
By changing the following variables and parameters asfollowsrrCCiL1 =Vd x , iL2 =Vd y ,L1L1pdv = Vd z , t = L 1 C τ , “ · ” =,(1)
rr dτ
L1L1C
β=, γ=g, δ = rsL2C
L1
where g is a linear negative conductance value of NR if
we consider 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 ©
ªẏ = β z − f (y)(2)
ż = −(x + y) − h(z)wheref (y) =o1n|δy + 1| − |δy − 1|2(3)andγ∗ n2KX©
ªo (4)(mk −mk+1 ) |z −pk+1 |−|z+pk+1 |h(z) = m0 γ ∗ z +k=0

Figure 3: Attractor drawing onto the z − x plane for theparameters β = 10.0, γ ∗ = 0.78 and δ = 100

Figure 4: A coupled network model which each circuit isconnected to four neighbors
f (y) is a function of the current y and h(z) is a functionof the voltage z, respectively. The function h(z) which isdesigned for several segment piecewise linear as symmetricwith respect to the origin. The parameter γ ∗ is used for abasic common value, hence the values mk (k = 0, 1, 2, · · ·,K) mean magnitude of the slope to the ratio for γ ∗ .
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
can confirm that chaotic and two periodic attractors coexist
in the circuit. This means coexistence of both chaos andtwo different size of limit cycles in the same parameters.
3. Simulation for coupled MSCOs
In this section, the model of MSCOs coupled by inductors as a non-uniform network system are investigated. For- 223 -example of a coupled network, now let us consider the coupled MSCOs model which combined number of N × M
chaotic circuits are connected by inductors to neighbors’
circuit as a network structure shown in Fig. 4. In this study,
the coupling strength of the network is a non-uniform that
it is different of the coupling strength between chaotic oscillators in the previous study [12]. The circuit index isdefined as (i, j). It is note that every chaotic circuit is composed by all the same parameters and connected to fourneighbors 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 thesimilar way described before, the circuit equations of coupled MSCOs can be normalized by the variables (1) witha new basic parameter α = L1 /L0 . Here, as we consider
a non-uniform network, we can change the parameter α to∗α(i,j). Therefore, the whole circuit equations can be rewritten as follows.
ẋ(i,j)ẏ(i,j)
 ż(i,j)







= z(i,j)¡¢= β z(i,j) − f (y(i,j) )³w=α(i,j)(x(i,j−1) − x(i,j))n+α(i,j)(x(i+1,j) − x(i,j))e+α(i,j) (x(i,j+1) − x(i,j))

4. Conclusions
In this study, we have investigated several synchronization modes in coupled multi-state chaotic oscillators ona non-uniform network system. Coexistence of several
types oscillation modes have been confirmed in the coupled
MSCOs. Several interesting chaotic phenomena of spatiotemporal behavior have been observed in the coupled network system. On a large scale of coupled chaotic 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.
Acknowledgment
This work was partly supported by The Hirao Taro Foundation of the Konan University Association for Academic
Research, Japan.
References(5)´
s+α(i,j)(x(i−1,j) − x(i,j))−(x(i,j) + y(i,j) ) − h(z(i,j) )where functions f (y) and h(z) are similar to (3) and (4),∗respectively. The parameter α(i,j)
is the coupling strengthof between both oscillators. The symbol ∗ means a target
which corresponds to left(=w), upper(=n), right(=e) andlower(=s), respectively.
We show some computer calculation results by using 4th order Runge–Kutta method with time step size ∆t =0.001 for the circuit equations in some cases of (N, M )as follows. The parameters of each circuit are the same in
the Sec. 2. The initial conditions of each circuit are given
at random. Figure 5 shows some typical results obtained
from computer simulation in some cases of coupled number (N, M ). Figure shows both uniform and non-uniform
network of coupling strength. The matrices as shown inFig. 6 also show all parameters of coupling strength α
which was made to be added a uniform random distributed
number of a section [−0.3, 0.3].
We can confirm complex and chaotic synchronization
phenomena on the coupled system. Because it cannot
be shown all simulation results, we can several types ofcomplex and interesting synchronization phenomena inthe same parameters, i.e., in-phase synchronization, antiphase synchronization, clustering of phase synchronization, phase locking and other types. Several phase synchronization modes are coexisting in spite of the same parameters. In the camera-ready paper, we will show thatstochastic analysis for these several synchronization modes
are investigated.

[1] S.P.Datardina, D.A.Linkens, “Multimode Oscillations in Mutually
Coupled van der Pol type Oscillators with Fifth-Power Nonlinear
Characteristics,” IEEE Trans. Circuits Syst., vol. 25, no. 5, pp. 308–315, 1978.
[2] M.E.Yalçin, J.A.Suykens, J.P.Vandewalle, ”Cellular Neural Networks, Multi-Scroll Chaos and Synchronization,” World Scientific
Publishing, 2005.
[3] L.O.Chua, M.Komuro, T.Matsumoto, “The Double Scroll Family,”IEEE Trans. Circuits Syst.–I, vol. 33, no. 11, pp. 1072–1118, 1986.
[4] J.A.K.Suykens, J.Vandewalle, “Generation of n-Double Scrolls (n= 1,2,3,4,...),” IEEE Trans. Circuits Syst.–I, vol. 40, no. 11, pp.
861–867, 1993.
[5] M.Wada, Y.Nishio, A.Ushida, ”Chaotic Itinerancy Phenomena on
Coupled n-Double Scrolls Chaotic Circuits,” Proc. of ISCAS’99,
vol. 5, pp. 487–490, 1999.
[6] G.Zhong, K.Man, G.Chen, “A Systematic Approach to Generating
n-Scroll Attractors,” Int. J. of Bif. and Chaos, vol. 12, no. 12, pp.
2907–2915, 2002.
[7] M.E.Yalçin, J.A.Suykens, J.P.Vandewalle, “Families of Scroll Grid
Attractors,” Int. J. of Bif. and Chaos, vol. 12, no. 1, pp. 23–41,
2002.
[8] M.Wada, Y.Nishio, “Coexistence Phenomena of Chaotic and NonChaotic Oscillations in a Three-Dimensional Chaotic Circuits,”
Proc. NOLTA’02, vol. 2, pp. 869–872, 2002.
[9] M.Wada, Y.Nishio, “Several Phase Synchronization Modes in Coupled Multi-State Chaotic Circuits,” Proc. of NOLTA’04, pp. 535–538, 2003.
[10] M.Wada, Y.Nishio, “Coexistence of Several Phase Synchronization in a Ring of Coupled Multi-State Chaotic Oscillators,” Proc.
NOLTA’07, pp. 393–396, 2007.
[11] M.Wada, M.Tanaka, Y.Nishio, “Stochastic Analysis of Several
Synchronization Phenomena on Coupled Chaotic Oscillators,”
Proc. of The 40th ISCIE International Symposium on Stochastic
Systems Theory and Its Applications, pp. 294–299, Nov. 2008.
[12] M.Wada, Y.Nishio, “Spatio-Temporal Phenomena of Coupled
Multi-State Chaotic Circuits,” Proc. of International Workshop onVisions, Communications and Circuits, pp. 23-26, 2008.
[13] N.Inaba, T.Saito, S.Mori, “Chaotic Phenomena in a Circuit with a
Linear Negative Resistance and an Ideal Switch of Diode,” IEICE
Trans., vol. E70, no. 8, pp. 744–754, 1987.

- 224 -(a)(b)(c)(d)
Figure 5: Examples of simulation results obtained from coupled MSCOs for α = 0.50 (in case of the uniform network),
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) uniform network of (3,3), (b) non-uniform network of (3,3), (c) uniform
network of (4,4) and (d) non-uniform network of (4,4) with waveform of each oscillator



0.303
0.748
 (1, 1) 0.415 (1, 2)
0.485
 0.527 (2, 1) 0.608 (2, 2)
 0.4230.711(3, 1) 0.343 (3, 2)



0.346
(1, 3) 0.222 

0.777


0.227 (2, 3) 0.683 
0.787
0.765 (3, 3) 0.457
0.705

0.34
 (1, 1)
 0.76 (2, 1)
 0.52
 (3, 1)
 0.70(4, 1)

0.49
0.42
0.59

0.38
(1, 2)
0.62(2, 2)
0.75(3, 2)
0.37(4, 2)

0.69
0.54
0.600.60(1, 3)
0.54(2, 3)
0.800(3, 3)
0.53(4, 3)

0.52
0.73
0.45

0.37
(1, 4)
0.50(2, 4)
0.78(3, 4)
0.46(4, 4)0.64 

0.46 

0.77 



0.42
0.20
0.73
0.40
(a)
(b)
Figure 6: Matrices of the coupling strength for (a) 3×3 and (b) 4×4 in the simulation results of Figure 5- 225 -