IEEE Workshop on Nonlinear Circuit Networks
December 14-15, 2012

Synchronization of Oscillatory Ring Topology
Coupled by Two Types of Resistors
Takahiro Nagai, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {nagataka, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization phenomena observed in coupled oscillatory networks by using twotypes of resistors. Negative and normal resistors are used ascoupling resistors in our circuit model. By giving particularparameters, we can observe the oscillation death and amplification of the amplitude. Finally we extend coupled oscillators topolygonal oscillatory networks.

(a)I. I NTRODUCTION
There are a lot of synchronization phenomena in this world.
This is one of the nonlinear phenomena that we can often
observe by natural beings which does collective actions. For
example, firefly luminescence, cry of birds and frogs, applauseof many people and so on. Synchronization phenomena havea feature that the set of small power can produce very
big power by synchronizing at a time. Therefore studies ofsynchronization phenomena have been widely reported not
only engineering but also the physical and biological fields [1][6]. Investigation of coupled oscillators attracts attention frommany researchers because coupled oscillatory network produce
interesting phase synchronization such as the phase propagation wave, clustering and complex patterns.
In our investigations, we use several van der Pol oscillators(see Fig. 1(a)). Van der Pol oscillators have been coupledin various form and investigated about their synchronization
phenomena [7], [8]. We have observed three-phase synchronization in the circuit model which are shown in Fig. 1(b) since
the discrepancy occurs in synchronization phenomena between
adjacent oscillators [9]. This three-phase synchronization isalways observed stability.
On the other hand, we can observe generically in-phasesynchronization in two coupled oscillators for Fig. 2(a) andFig. 1.

(b)Fig. 2. Synchronization phenomena between first and second oscillators.
(a) In-phase synchronization. (b) Anti-phase synchronization.

anti-phase synchronization in the case of Fig. 2(b). Namely
what kind of synchronization phenomena will be observedin coupled oscillators which are included these two types ofsynchronization states? First, we consider coupled oscillators
including two types of coupling resistor which is shown in
Fig. 3. Seeing partly, anti-phase synchronization is observed
between first and second oscillators and in-phase synchronizations are observed between first and third or second andthird oscillators. However because frustration occurs in wholesystem, we guess unexpected synchronization phenomena arefound out.
II. S YNCHRONIZATION P HENOMENA
In this study, we investigate synchronization phenomena of
coupled oscillatory networks using different types of resistors(a) Van der Pol oscillator. (b) Three coupled oscillators.

Fig. 3.

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Conceptual circuit model for three coupled oscillators.

(a)

(a)

(b)

(b)

(c)

(d)

Fig. 4. Several types of coupled polygonal oscillatory networks. (a) tetragon
network. (b) pentagon network. (c) hexagon network. (d) heptagon network.

(c)Fig. 5. Simulation results for polygonal oscillatory networks. (a) triangle
network for γ1 = γ2 = 0.050. (b) pentagon network for γ1 = γ2 = 0.050.
(c) heptagon network for γ1 = γ2 = 0.075.

partially. We focus on the amplitude of each oscillator andphase differences between adjacent oscillators by changing
coupling strength. Also, we increase number of oscillators in
the ring topology and we consider differences of synchronization phenomena by changing number of oscillators.

synchronization state between first and second oscillators isobserved anti-phase synchronization and case of between otheroscillators are observed in-phase synchronization except thefarthest oscillator.

III. S YNCHRONIZATION P HENOMENA

IV. C ONCLUSION

In this section, we show the simulation result for several
types of oscillatory networks as shown Fig. 4. We calculate
circuit equations using the fourth-order Runge-Kutta methodwith the step size h = 0.001. We fix the parameter ε = 0.020
in these simulations. First, we explain the synchronization phenomena in the case of triangle networks for coupling strength
γ1 = γ2 = 0.050. We show the simulation result of eachoscillator in Fig. 5(a). Obvious difference of the amplitude ofeach oscillator appears in this parameters. Namely, oscillation
death is observed in third oscillator. Phase differences between
first and second oscillators shows anti-phase synchronization,
because third oscillator stops oscillating in order to match
consistency.
In the case of polygonal oscillatory networks, different
synchronization phenomena for odd and even number are
produced even if same parameter are given. For example,
giving same values for coupling strength, odd polygonal
oscillatory networks cause oscillation death. The oscillator
which oscillation stops is the farthest oscillator from first andsecond oscillators. We show the farthest oscillator as blue
oscillator in Fig. 4(b) and (d). Also, getting away first andsecond oscillators, oscillators attenuate. Figure 5(b) and (c)show simulation results for odd polygonal networks and we
set coupling strength γ1 = γ2 = 0.050 for pentagon networkand γ1 = γ2 = 0.075 for heptagon network. In the case ofheptagon network, fifth oscillator attenuates but does not occur
the oscillation death for γ1 = γ2 = 0.050.
On the other hand, even polygonal oscillator networks can
not produce oscillation death but we can observe small andlarge of amplitudes. Regardless of the number of oscillators,

In this study, we have been able to observe several patternsof synchronization phenomena. For three coupled oscillators,
we have observe the oscillation death in third oscillator. Inaddition, we have investigated differences of synchronization
phenomena in the ring topology by increasing number ofoscillators. We could confirm that odd polygonal networks
produced oscillation death but even one do not produce
oscillation death.
In our future works, we will derive each amplitude forpolygonal networks by using averaging method.
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