2004 International Symposium on Nonlinear
Theory and its Applications (NOLTA2004)
Fukuoka, Japan, Nov. 29 - Dec. 3, 2004

Several Phase Synchronization Modesin Coupled Multi-State Chaotic Circuits
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—Nonlinear dynamics on coupled chaotic oscillatiors is considerable interesting for a wide variety ofsystems in several scientific fields and applications. Thispaper presents a novel type of several phase synchronization modes in coupled asynchronous multimode chaoticoscillators. Each chaotic circuit can individually behaveboth chaotic or periodic oscillations in the same parameters
asynchronously. In this study, such chaotic circuits coupled
by some inductors as a ring are proposed and classificationsof phase synchronization modes are investigated. In numerical simulation, many types of phase synchronization
modes are confirmed.
1. Introduction
Many types of coupled systems have been widely studied
in order to clarify inherent features and many researchers
have already proposed and investigated them. Coupledchaotic 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 on several coupled systems is still considerable interest from the viewpoint of both natural scientific fields
and several applications. They have been confirmed in several systems; e.g., coupled van der Pol oscillators[1], laser
systems[2], and so on. 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 theactive elements including in the systems have complexity constructed by compound some nonlinear elements,
it can be easily consider that they yield several interesting features. The circuit which can individually behaveboth chaotic or periodic oscillations in the same parameters had been shown[8]. This type of circuit was called amulti–state chaotic circuit(abbr. MSCC). Multimode oscillations in coupled two multi–state chaotic circuits had also
been investigated[9]. Such complex and strange nonlinear structures yield a wide variety of chaotic phenomena.
It is known that complex behavior can be confirmed suchchaotic itinerancy and spatio-temporal chaos on the large
scale coupled networks.
In this study, a novel type of phase pattern or multi-L0vk

L0C
Ck
L0

L0vN

L0

C
CNv1
C
C1

L0v2
C
C2

Figure 1: Coupled model of chaotic circuits by several inductors as a ring.

mode asynchronous oscillations on the coupled MSCCs asa ring is investigated. The schematic diagram of the coupled circuits is shown in Fig. 1. In the past our works, wehad presented and confirmed that multi-state oscillations
both chaotic and non–chaotic (limit cycle) can be generated asynchronously in the same parameters on the computer simulations [8] and also that realization on the realcircuits [10]. In this paper, firstly the design scheme of aMSCC is shown briefly. Secondary both chaotic and periodic oscillations in the same parameters which can be confirmed in numerical simulations and circuit experiment arealso shown. Finally phase synchronization and classification of several phase patterns in some MSCCs coupled byinductors 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. 2 is modified chaotic circuitfrom a change model in a well–known three dimensional
chaotic circuit proposed by Inaba and Saito[11]. The original circuit consists of three memory elements, some diodes
and designed negative resistors. It is well known that it
can behave as Rössler type chaotic motions. We substitute a symmetrical continuous five segments piecewise lin-525

f (yk ) =Figure 2: Proposed chaotic circuit with five–segment piecewise linear resistors.
ear resistor for the negative active resistor including in theoriginal chaotic circuit. Further this circuit possesses another symmetrical piecewise 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
piecewise linear functions vd (iL2k ).
vd (iL2k ) =1(rs iL2k + Vd − |rs iL2k − Vd |)2(1)where threshold voltage Vd is realized by the total ofthreshold of the diodes and supply DC voltage E1 , and rs
is a resistance value at the parallel diode in while off state.
The variable vd (iL2k ) determines their chaotic dynamics.
We now consider the coupled model which combined N
chaotic circuits are connected by inductors L0 as a ringstructure. The chaotic circuits are composed by all thesame parameters. Therefore when we choose a thresholdvoltage value Vd as a criterion, the circuit equation of coupled MSCCs can be normalized by changing the following
variables and parameters,rr
CCiL1k =Vd xk , iL2k =V d yk ,L1L1pd,
vk = Vd zk , t = L1 C dτ , “ · ” =(2)dτrr
L1L1C
L1α=, β=, γ=g, δ = rsL0L2C
L1
where g is a temporary parameter as a linear negative conductance value of NR if we consider the negative resistor
as an ideal linear function. The differential equation concerned with L0 can be eliminated by using KCL at the loopof the inductors L0 and L1k , then the circuit equations ineach system are reduced to three–dimensional equations.
Consider that the part of negative resistance NR in Fig. 2
replaces to the function h(zk ) of a voltage source zk , thenthe circuit equations can be rewritten byx˙k = zk¡¢y˙k = β zk − f (yk )(3)z˙k = α(xk−1 − 2xk + xk+1 )−(xk + yk ) − h(zk )o1n|δyk + 1| − |δyk − 1| .
2(4)The function h(z) which can be designed by symmetrical five–segment piecewise linear with respect to the origin
for the parameters four breakpoints at {±Bp1 , ±Bp2 } andfive slopes by {m0 , m1 , m2 , m1 , m0 } is described with acanonical form as follows.
¡¢γ∗ nh(z) , m0 γ ∗ z+
(m0 −m1 ) |z−Bp2 |−|z+Bp2 |2¡¢o (5)+(m1 −m2 ) |z−Bp1 |−|z+Bp1 |where the parameter γ ∗ is used for a basic value, hence thevalues mk (k = 0,1,2) mean the ratio to the value γ ∗ .
We can realize the MSCC on the real circuit. In orderto realize the nonlinear characteristic of NR , we designed
a piecewise linear resistor constructed by using some operational amplifiers(op amps) and resistors. The details of aconstruction are described in [12].
Our proposed circuit can behave both chaotic and periodic oscillations in the same parameters when we supply with different initial conditions. Figure 3 shows somesnapshots obtained from circuit experiment. The detailed
schematic design had been explained in Ref. [8][9][10] andthe circuit settings are put in the caption. As a result, bothchaotic and periodic attractors can be observed in the same
circuit parameters.
Figure 4 also shows a typical chaotic attractor obtained
from computer calculation results in the case of N = 1 forthe parameters β = 10.0, γ ∗ = 0.78, δ = 100, with piecewise linear characteristics realized by breakpoints Bp1 =0.30, Bp2 = 0.56, slopes m0 = −1.0, m1 = 0.65 andm2 = −0.2. We can confirm that both chaotic and periodic attractors coexist in the same parameters. In order toknow structure of these attractor, the 3-D shape attractor
is also drawn. Further a bifurcation diagram by changing
the parameter γ ∗ is shown in Fig. 5. As increasing γ ∗ periodic attractor bifurcates to chaos in the following routes
while keeping the limit cycle at around the origin. Oscillation of symmetrical 1-period → asymmetrical 1-period→ bifurcates to 2n period → asymmetrical slight chaos →symmetrical fluttered chaos. We can observe that both twooscillation modes exist separately in the same parameters.
However, chaotic attractor disappear and limit cycle is only
observed when γ ∗ is larger than around 0.876(≡ γ ∞ ). The
reason is that circuit dynamics will stabilize and settle in alimit cycle if trajectory becomes larger and is soon drawn
into inside area.
3. Numerical Simulations
In this section, the model of coupled MSCCs by inductors are investigated. We show some computer calculation
results by using 4-th order Runge–Kutta method with time
step size ∆t = 0.001 for the circuit equation (3), (4) and(5) in some cases of N = 2 ∼ 7 as follows.

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3.1. Two subcircuits case N = 2i6-v
(a)

(b)
(c)
Figure 3: Some snapshots of the circuit experiment and numerical simulation. (a) v–i characteristics of the designed
piecewise linear resistor, horizontal: 5V/div. (b) chaotic attractor, (c) limit cycle, horizontal: 2V/div. Circuit parameters settings: L1 = 123.1mH, L2 = 10.2mH, C = 68.7nF,
R1 = 33.2kΩ, R2 = 21.7kΩ, R3 = 1.22kΩ, R4 = 196Ω, R5 =333Ω, R6 = 1.47kΩ, R7 = 10.3kΩ, E1 = 4.80V, E2 = 2.78V.
Threshold voltage of one diode vth ' 0.78V.

(a)(b)
Figure 4: Drawing attractor onto the z − x plane in (a),and 3-D trajectories in (b) for the parameters β = 10.0,
γ ∗ = 0.78 and δ = 100. h(z): [Bp1 , Bp2 , m0 , m1 , m2 ] =[0.35, 0.55, -1.0, 0.65, -0.20]

Now we consider that the number of the coupled MSCCs
is two. This case corresponds to the model in [9]. Though
the detail results are omitted, we can confirm several typesof phase synchronization modes in this model. In this case,four asynchronous oscillation modes could be confirmed
consequently by numerical simulations when the initialconditions are varied. We could observe an in-phase synchronous limit cycle, an anti-phase synchronous limit cycle, an anti-phase chaotic synchronous state, and a doublemode oscillations in all the same parameters. In this coupled two MSCCs, double–mode chaotic oscillations wereconfirmed.
3.2. Three 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 6(a) shows a case of threephase synchronization of three limit cycles in while keeping 2π/3 phase difference. From top of the figure, attractors drawing onto z–x plane, synchronization state of zk –zk+1 plane, and waveform of difference between the twovariables zk − zk+1 . Figure 6(b) shows in-phase synchronization of them. They are corresponding normally to threephase synchronization in generic oscillators. The figure (c)shows a multimode oscillation of both chaotic and periodic
attractors. Further (d) shows a new type of synchronization mode applicable to no other one. We could confirm tocoexist with several types of synchronization modes.
3.3. Discussion of Coupled MSCCs N >=4
In large coupled systems for N >= 4, it is easily expected
to be confirmed more complex behavior. We now showonly some results in Fig. 7 for the case of N = 4 and 7. In
the case of N = 4, several types of synchronization modes
are confirmed. On the other hand, in the case of N = 7,
no much many synchronization modes have confirmed insuch parameter settings. However when the number of N
is large and the circuit parameters should be set appropriately, a certain kind of phase propagation phenomena may
be confirmed. Furthermore several complex behavior could
be also confirmed in a large number of N .
4. Conclusions

Figure 5: Bifurcation diagram by changing the parameterγ ∗ from 0.2 to 1.0 for β = 10.0 and δ = 100.

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, we con-527

(a)

(b)

(c)
(d)
Figure 6: Some simulation results in the case of three MSCCs coupled by inductors. α = 0.50, β = 10.0, γ ∗ = 0.68,
δ = 100. h(z): [Bp1 , Bp2 , m0 , m1 , m2 ] = [0.35, 0.55, -1.0, 0.65, -0.20]. (a) three phase synchronization, (b) inphase synchronization, (c) multimode synchronization of both chaotic and periodic oscillations and (d) other type ofsynchronization modes.
sider that several types of complex behavior are expected
to yield novel chaotic phenomena e.g., chaotic itinerancy,
spatio–temporal chaos, multi–agent systems, soliton likewave propagation phenomena, and inherent emergent property, in which concerned with other current topics.

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Figure 7: For some cases of three MSCCs coupled by inductors. The parameters are given by all the same settings
as in Fig. 6 except for the number of circuits.

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