Proceedings of 2015 IEEE Workshop on Nonlinear Signal Processing (NSP2015)
Tokushima University, Tokushima, Japan. December 25, 2015

Synchronization in Complex Networks Contain Dispersion
by Using Parametric Excited Oscillators
Kosuke Oi, Yoko Uwate and Yoshifumi Nishio(Tokushima University)
Email: {ooi, uwate, nishio}@ee.tokushima-u.ac.jp

S UMMARY
Synchronization is one of the fundamental phenomena in nature and it is observed over the various fields. Studies on
synchronization phenomena of coupled oscillators are extensively carried out in various fields, physics [1], biology [2],engineering and so on. The coupled van der Pol oscillator is one of coupled oscillators, and synchronization generated in
the system can model certain synchronization of natural rhythm phenomena. Parametric excitation circuit is one of resonant
circuits, and it is important to investigate various nonlinear phenomena of the parametric excitation circuits for future engineering
applications. In simple oscillator including parametric excitation, Ref. [3] reports that the almost periodic oscillation occurs if
nonlinear inductor has saturation characteristic. Additionally the occurrence of chaos is referenced in Refs. [4] and [5].
In our research group, we have investigated synchronization of parametrically excited van der Pol oscillators [6]. By carryingout computer calculations for two or three subcircuits case, we have confirmed that various kinds of synchronization phenomenaof chaos are observed. In the case of two subcircuits, the anti-phase synchronization is observed. In the case of three subcircuits,
self-switching phenomenon of synchronization states is observed.
However, we have investigated the only simple network models. It is important to investigate more complex network forthe broad-ranging future engineering application. In this study, we challenge to investigate more complex network containdispersion using parametrically excited van der Pol oscillators. In our previous study, we have challenged to investigate thesynchronization and clustering in more complex network using ten coupled parametrically excited van der Pol oscillators withsmall mismatch [7].
In this study, we primarily intended to investigate effect of the hub in network of the synchronization. First, we compare
synchronization phenomena of parametrically excited van der Pol oscillators with small mismatch in complex network withhub and without hub. Next, we investigate the mechanism of full synchronization in random network by adding several typesof small mismatch to hub of random network.
R EFERENCES[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.3423-3433,
Nov. 2005.
[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of nonlinear electronic oscillators for neural computation,” IEEE Trans. NeuralNetworks, vol.15, no.5, pp.1315-1327, Sep. 2004.
[3] C. Hayashi, “Nonlinear Oscillations in Physical Systems,” Chap. 11, McGraw-Hill, New York (1964).
[4] C. Hayashi, M. Abe, K. Oshima and H. Kawakami, “The method of mapping as applied to the solution for certain types of nonlinear differential equations,”
Ninth International Conference on Nonlinear Oscillations, Kiev (Aug.-Sept.1981).
[5] M. Inoue, “A Method of Analysis for the Bifurcation of the Almost Periodic Oscillation and the Generation of Chaos in a Parametric Excitation Circuit,”
Trans. of IEICE, vol. J68-A, no. 7, pp. 621-626, 1985.
[6] H. Kumeno, Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proc. NOLTA’08, pp. 128-131, Sep.
2008.
[7] K. Oi, Y. Uwate, Y. Nishio, “Synchronization and Clustering in Coupled Parametrically Excited Oscillators with Small Mismatch,” Proc. ISCAS’15, pp.
910-913, May, 2015.

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