2008 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'08, Gold Coast, Australia, March 6-8, 2008

Phase Synchronization Phenomena on Coupled Multi-State Oscillators
Masahiro Wada† and Yoshifumi Nishio‡†Konan University
8-9-1 Okamoto, Higashinada, Kobe, Japan
Phone/Fax:+81-78-435-2531
Email: wada–m@konan–u.ac.jp

Abstract
This paper presents several phase synchronization phenomena on multi–state oscillators coupled by inductors as a ring.
Each oscillator circuit can individually behave some periodic oscillations (limit cycles) in the same parameters asynchronously. In this study, a coupled oscillator circuit which iscomposed by some oscillators is proposed and classificationof phase synchronization modes is investigated. In numericalsimulation, many types of phase synchronization modes are
confirmed in the proposed systems.
1. Introduction
Oscillator is an important device and one of essential component in the natural world. Nonlinear dynamics on coupledoscillators is considerable interesting for a wide variety ofsystems in several scientific fields and some engineering applications. 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 of multimode oscillations orphase synchronization on several coupled systems is still considerable interest from the viewpoint of both natural scientificfields and several applications.
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 forseveral 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 by compound some nonlinear elements, it can beeasily 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]-[12]. This type of circuit
was called a multi–state chaotic circuit (abbr. MSCC). Multimode oscillations in coupled two or more multi–state chaotic‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: nishio@ee.tokushima-u.ac.jp

circuits had been shown [10] on real circuit. A complicatedand interesting phenomena of phase synchronization had also
been investigated [11][12]. It is known that complex behavior
can be confirmed such chaotic itinerancy and spatio–temporalchaos on the large scale coupled networks. Some kinds ofoscillation modes had been reported on large scale coupledchaotic circuits such phase synchronization, phase propagation and frustration of oscillation and so on [13]-[15]. On theother hand, the coupled van del Pol oscillators with hard nonlinearity had been investigated [16] and also stability analysisof them [17]. There are many oscillators from very low tovery high frequency, which can be easily constructed on thereal electrical circuits. We consider that it is important to investigate phase synchronization and pattern dynamics in suchcoupled oscillator systems.
This paper presents several phase synchronization modesof multi–state oscillator circuits (abbr. MSOC) coupled byinductors as a ring. This is a kind of val der Pol type oscillator and a typical two dimensional autonomous system, which
consists of two memory elements and a designed piecewise
linear resistor. We substitute a symmetrical continuous segments piecewise linear resistor for the negative active resistor
including in the original chaotic circuit. Each circuit can individually behave some periodic oscillations (limit cycles) in
the same parameters asynchronously. This proposed circuit
can behave three limit cycles when we supply with different
initial conditions. In this study, a coupled system which isconnected to some oscillators is proposed and classificationof phase synchronization modes is investigated. 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.

2. Model Description
The circuit shown in Fig. 1 is an oscillator circuit of a well–known two dimensional van der Pol type oscillator. The original circuit consists of two memory elements and a designed
negative resistor.

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L0vkL0i+v

OCkLNROCC-rL0L0vNO
CN

Figure 1: Based oscillator circuit with piecewise linear resistors NR .

v1
OC1L0v2
OC2

Figure 4: Concept of a model of oscillators coupled by inductors as a ring.

h(v)

½0L0v

Figure 2: Design of NR in the oscillator for sawtooth piecewise linear resistor with respect to the origin.

ẋ =ẏ =y−x − γh(y)h(y) = mN yN1X+(mk − mk−1 ) {|y − bk | − |y + bk |}
2(2)

(3)

k=1

where h(y) is the piecewise nonlinear resistor with respect tothe origin. The parameter γ is used for a basic common value,
hence the values mk ( k = 0, 1, 2, · · ·, N ) mean magnitude of
the slope to the ratio for γ, the values bk means break points.
Therefore, the piecewise linear function has 2N + 1 slopes
and N breakpoints.
For example, the circuit parameters in the case of N = 5
Figure 3: Three steady state of the oscillator in the same pa- are as follows.
rameters.
γ = 0.78
In this study, we substitute a symmetrical continuous piece{m0 , m1 , m2 , m3 , m4 , m5 }wise linear resistor for the negative active resistor including= {−0.5, 4.0, −2.0, 4.0, −2.0, 2.0}
(4)
in the circuit. It can be realized the segment-wise nonlinear
resistor which can be easily constructed by some subcircuits.
{b1 , b2 , b3 , b4 , b5 }The designed piecewise linear resistor is shown in Fig. 2. The= {0.2, 0.25, 0.4, 0.5, 0.7}
piecewise linear resistor can be easily constructed by combining some components in parallel [9][10]. Let us consider the This case corresponds to 11-segments piecewise linear resiscircuit equations by changing the following variables and pa- tor as shown in Fig. 2.
Figure 3 shows some computer simulation results in therameters.
case of (4) when the parameters are all the same and the initial conditions of each circuit are only different. In these par√Crameters, we can confirm that three limit cycles coexist in thex, v = y, t = LCτ, “ · ” = d/dτ,i=Lsame parameters as a steady state. Hereafter we use these(1)rparameters (4) in numerical simulation.
Lγ=g, and r ' 0C
3. Simulation for coupled MSOCs
where g is a linear negative conductance value of NR if we
assume the negative resistor is as an ideal linear. Replace
In this section, the model of coupled MSOCs by inductors
the part of negative value NR in Fig. 1 to the function h(y) are investigated. We now consider the coupled model which
represented by a voltage source y as a canonical form. Then oscillator circuits to the number of N are connected by inthe normalized circuit equations can be rewritten as follows. ductors L0 as a ring structure as shown in Fig. 4. All circuit- 145 -pare with the case N = 2, several different synchronization
phenomena can be found. Because all types of the results can
not be represented, some simulation results are only shown
here. Figures 5 and 6 show some computer simulation results
obtained by the circuit parameters (4) and α = 0.50. From
top of the figure, attractors drawing onto y–x plane, synchronization state for both of phase yk − yk+1 and xk − xk+1 ,and waveform of each oscillator, which k obeys in the orderof cyclic rule for any number of coupled oscillators. Figure 5 shows some simulation results in the case of N = 3;(a) in-phase synchronization of three limit cycles, (b) threephase synchronization, (c) double-mode oscillation, and (d)phase locking phenomena. We could confirm to coexist a lotof synchronization modes in this situation.
In a large number of coupled systems for N ≥ 4, it is easily
expected to be confirmed more complex behavior. We show
some results in Fig. 6 for the case of N = 4 and 6; (a) in-phasesynchronization, (b) two-pair of in-phase and anti-phase synchronization of limit cycles, (c) two-pair of synchronization
with phase difference, (d) phase locking phenomena, respectively for N = 4. On the other hand, in the case of N =6,
several phase synchronization modes had been confirmed, forexamples as shown in Fig. 6(e)–(g). Thus, several complex
behavior and phase synchronization phenomena could be also
confirmed in the coupled multi–state oscillators in the sameparameters.

(a)

(b)

(c)

(d)

4. Conclusions
Figure 5: Some simulation results in the same parameters forthe case N = 3.
parameters are set in the same values. The circuit equation of
coupled MSOCs can be normalized by changing the variables(1) and a new parameter α = L/L0 as coupling strength, thenthe circuit equations can be rewritten as follows.
 x˙ky˙ k==ykα(xk−1 − 2xk + xk+1 )−xk − γh(yk )(5)Here, we show some computer simulation results by using
4-th order Runge–Kutta method with time step size ∆t =0.001 for the circuit equation (5) and (3) in some cases of
N = 2 ∼ 6 as follows.
First, we consider that the number of the coupled MSOCs
is two. In this case, some asynchronous oscillation modes
could be confirmed consequently by numerical simulations
when the initial conditions are changed. We can confirm several types of phase synchronization phenomena in this modelfor their size of the limit cycles; in-phase synchronous limit
cycles, anti-phase synchronous limit cycles, and multimodeoscillations, in the same parameters, respectively.
Second, we consider the case of N = 3 or more. Com-In this study, we have investigated several synchronization
modes in coupled multi–state oscillator circuits. Coexistenceof several types oscillation modes have been confirmed incoupled MSOCs by inductors as a ring for several cases. On
large scale coupled oscillator including chaotic circuits sucha scale-free network, we consider that several types of complex behavior are expected to yield novel complicated phenomena e.g., spatio–temporal behavior, multi–agent systems,
soliton like wave propagation phenomena, and inherent emergent property, in which concerned with other current topics.
Furthermore, it is also an important work to analyze stabilityof the multi–state oscillator.
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(g)

Figure 6: Some simulation results in the same parameters forthe cases N = 4 and N = 6.

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