2007 International Symposium on Nonlinear Theory and itsApplications
NOLTA'07, Vancouver, Canada, September 16-19, 2007

Coexistence of Several Phase Synchronization
in a Ring of Coupled 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
by inductors as a ring. Several phase patters occured insome types of coupled oscillators have been widely known.
Each chaotic circuit which used in this paper can individually behave both chaotic and periodic oscillations in thesame parameters asynchronously. In this study, such thecoupled chaotic circuits are proposed and classifications ofphase synchronization modes are investigated. In numerical simulation, many types of phase synchronization modes
are confirmed.
1. Introduction
Nonlinear dynamics on coupled chaotic oscillators isconsiderable 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
of chaotic multimode oscillations or chaotic itinerancy onseveral coupled systems is still considerable interest fromthe viewpoint of both natural scientific fields and severalapplications. They have been confirmed in several systems; e.g., coupled van der Pol oscillators [1], laser systems[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 in detail. 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 circuit(abbr. MSCC). Multimode oscillations in coupled two or
more multi–state chaotic circuits had also been investigated[10][11]. Such complex and strange nonlinear structures
yield a wide variety of chaotic phenomena. It is an important idea that how does nonlinearity lead to various kindsof behaviors. They are also constructed by making several
equilibrium points. A mechanism and search of chaotic
regions in piecewise linear systems were investigated [9].
The purpose of our study is to clarify coexistence of bothchaotic and non–chaotic behavior in the chaotic systems.
Further complex behavior in the large scale network of thecoupled chaotic circuits are also investigated. It is known
that complex behavior can be confirmed such chaotic itinerancy and spatio-temporal chaos on the large scale coupled networks. Some kinds of oscillation modes had beenreported on large scale coupled chaotic circuits such phasesynchronization, phase propagation and frustration of oscillation and so on [12]-[14].
In this study, several phase patterns and multimode asynchronous oscillations on the coupled MSCCs are investigated. There is a typical three dimensional autonomouschaotic system, which consists of three memory elements,
some diodes and designed negative resistors. It is wellknown that it can behave as Rössler type chaotic motion.
We substitute a symmetrical continuous segments piecewise linear resistor for the negative active resistor includingin the original chaotic circuit. This proposed circuit can behave both chaotic and periodic oscillations in the same parameters when we supply with different initial conditions.
In this paper, phase synchronization and classification ofseveral phase patterns in some MSCCs coupled by inductors as a ring are investigated. Several types of phase synchronization modes are confirmed asynchronously, but all
circuit parameters are the same.

2. Model Description
The circuit shown in Fig. 1 is modified chaotic circuitfrom a well–known three dimensional chaotic circuit [15].
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 motion.
We substitute a symmetrical piecewise linear resistor forthe negative active resistor including in the original chaotic
circuit. Further this circuit possesses another symmetricalpiecewise nonlinear resistor with respect to the origin.
At first, we approximate the i − v characteristics in thepart of both diodes and E1 by the following three-segment- 393 -

i L1

VdL2+v

vdi L2NRrd / 2 = r sL1C-D

iof the diodes, then the circuit equations can be rewritten by ẋ = z ¡¢(3)ẏ = β z − f (y) ż = −(x + y) − h(z)f (y) =-Vd

Figure 1: Based chaotic circuit with piecewise linear resistors.

γ∗ n2KX©
ªo(mk −mk+1 ) |z−pk+1 |−|z +pk+1 |h(z) = m0 γ ∗ z+m0
m2

m3
m4-p3-p2m3

m1

p4-p40

m0

p2
p3p1v

Figure 2: Design for sawtooth nonlinear resistor with respect to the origin.
piecewise linear functions vd (iL2k ).
vd (iL2k ) =(5)k=0m2-p1(4)andh(v)m1ª1©
|δy + 1| − |δy − 1| ,21
(rs iL2k + Vd − |rs iL2k − Vd |)2(1)where threshold voltage Vd is realized by the total ofthreshold of the diodes and rs is a resistance value at theparallel diode in while off state. The variable vd (iL2k ) determines their chaotic dynamics.
In this study, we substitute a symmetrical continuous
piecewise linear resistor for the negative active resistor including in the chaotic circuit. The designed chaotic circuit
is shown in Fig. 2. The piecewise linear resistor can be easily constructed by combining some components in parallel[8][10]. By changing the following variables and parameters as followsr
CC
Vd x , iL2 =Vd y ,iL1 =L1
L1
pdv = Vd z , t = L1 C dτ , “ · ” =,dτrr
L1L1C
β=, γ=g, δ = rs
L2CL1r(2)

where g is a linear negative conductance value of NR ifwe consider the negative resistor as an ideal. Consider thatthe part of negative resistance NR in Fig. 1 replaces to thefunction h(z) represented by a voltage source z as canonical form. When we chose Vd for a threshold voltage value

where f (y) is a function of the current y and h(z) is afunction 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 ∗ isused for a basic common value, hence the values mk (k =0,1,2,· · ·,K) mean magnitude of the slope to the ratio forγ∗.
The detailed schematic design how to construct had been
explained and the circuit settings are put in their figures’caption [8]-[11]. As a result, both chaotic and periodic attractors can be observed in the same circuit parameters. In
the previous work, we could confirm that both chaotic andnon-chaotic attractors coexist in the same parameters. If
the sawtooth nonlinear resistor in the circuit was improved
to have several segments, we can confirm coexistence ofseveral attractors in the same parameters. Figure 3 also
shows a typical chaotic attractor obtained for the parameters β = 10.0, γ ∗ = 0.78, δ = 100, with piecewise 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 both chaotic and two periodic attractors coexist in thesame parameters. It is different from the previous work that
we could confirm coexistence of chaotic attractor and twodifferent size of limit cycles in this circuit.
3. Simulation for coupled MSCCs
We now consider the coupled model which combined
number of N chaotic circuits are connected by inductors
L0 as a ring structure. The chaotic circuits are composedby all the same parameters. Therefore when we choose athreshold voltage value Vd as a criterion, the circuit equation of coupled MSCCs can be normalized by changing thevariables (2) and a new parameter α = L1 /L0 , then theentire circuit equations can be rewritten byx˙k = zk¡¢y˙k = β zk − f (yk )(6) z˙k = α(xk−1 − 2xk + xk+1 )−(xk + yk ) − h(zk )In this section, the model of coupled MSCCs by inductors are investigated. We show some computer calculation- 394 -

Figure 3: Drawing attractor onto the z −x plane and 3-D trajectories for the parameters 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}.
results by using 4-th order Runge–Kutta method with time
step size ∆t = 0.001 for the circuit equation (6), (4) and(5) in some cases of N = 2 ∼ 7 as follows.

L0vk

L0C
Ck

3.1. Two subcircuits case N = 2
Now we consider that the number of the coupled MSCCs
is two. This case is similar to the model in the sense of coexistence of chaotic and periodic oscillations for the previous work [10]. Although the detail results are omitted, wecan confirm several types of phase synchronization modesin this model. In this case, some asynchronous oscillation
modes could be confirmed consequently by numerical simulations when the initial conditions are varied. We could
observe two different limit cycles by their oscillation size;
in-phase synchronous limit cycles, anti-phase synchronous
limit cycles, anti-phase chaotic synchronous state, and multimode oscillations in the same parameters.
3.2. Three or more subcircuits case N ≥ 3
In this section, we consider the case of N = 3. The
circuit parameters in each MSCC are set as all the same
parameters in the section 2 with additional parameter α =0.50. Compare with the case N = 2, several different synchronization phenomena can be found. Because all typesof the results can not be represented, some simulation results are only shown here. Figure 5(a) shows a case of inphase synchronization of three limit cycles. From top of
the figure, attractors drawing onto z–x plane, synchronization state of zk –zk+1 plane, and waveform of difference
between the two variables zk − zk+1 . Figure 5(b) shows inphase chaotic synchronization and pair of double-mode oscillation. They are corresponding normally to three phasesynchronization in generic oscillators. We could confirm tocoexist a lot of synchronization modes.
In large coupled systems for N ≥ 4, it is easily expected
to be confirmed more complex behavior. We now show
only some results in Fig. 5 for the case of N = 4 and 7. In
the case of N = 4, several types of synchronization modes
are confirmed. Figure (c) shows two-pair of in-phase andanti-phase synchronization of limit cycles. Further, figure(d) shows a new type of synchronization mode applicable

L0L0vNCCNL0v1
CC1

L0

v2
C
C2

Figure 4: Coupled model of chaotic circuits by several inductors as a ring.

to no other one. In the case of N = 7, no much many
synchronization modes have confirmed in such parametersettings. Figures (e) and (f) show typical simulation results
for a large number of N . However when the number of Nis large and the circuit parameters should be set appropriately, a certain kind of phase propagation phenomena may
be confirmed. Thus, several complex behavior could bealso confirmed in the coupled multi–state chaotic oscillators.

4. Conclusions
In this study, we have investigated several synchronization modes in coupled multi–state chaotic circuits. Coexistence of several types oscillation modes have been confirmed in coupled MSCCs by inductors as a ring for several cases. On large scale coupled chaotic circuits such ascale-free network, we consider that several types of complex behavior are expected to yield novel chaotic phenomena e.g., chaotic itinerancy, spatio–temporal chaos, multi–agent systems, soliton like wave propagation phenomena,and inherent emergent property, in which concerned withother current topics.

- 395 -(a)

(b)

(c)

(d)

(e)
(f)
Figure 5: Some simulation results obtained from coupled three, four and seven MSCCs for α = 0.50, 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}.
(a) in-phase synchronization, (b) double-mode and one-pair synchronization, (c) two-pari anti-phase synchronization, (d)two-pair double-mode and anti-phase synchronization, (e) seven-phase synchronization, and (f) multimode oscillation.
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