IEEE Workshop on Nonlinear Signal Processing
December 26, 2016

Synchronization Phenomena and Oscillation Death in Star-Coupled van der Pol Oscillators
by Adding Different Frequency Oscillators
Tran Minh Hai, Kosuke Oi, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {minhhai,oi,uwate,nishio}@ee.tokushima-u.ac.jp
Summary
Synchronization has grown to a considerable research field in recent years. Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems.
Studies of synchronization phenomena have been reported in many fields: electrical and electronic engineering, physics,biology, chemistry, etc. Furthermore, the applications of synchronization phenomena have also found in chemical, physicaland biological fields: about chaos synchronization of chemical models, about synchronization and coupling analysis of applied cardiovascular physics in sleep medicine, or about multistate and multistage synchronization with excitatory chemicaland electrical synapses, etc. In addition, in the organism, synchronization phenomena has been observed in the activity ofthe brain and the operation of the heart. It is considered to have become an important role. In the future, analysis of brain
activity and analysis of heart activity are very important. Its application to engineering fields such as the realization of brain
computer is expected.
The van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol,van der Pol found stable oscillations, which he called relaxation-oscillations, and are now known as limit cycles, in electricalcircuits employing vacuum tubes. In common, van der Pol oscillator is a non conservative oscillator with non-linear damping.
It evolves in time according to the second order differential equation:
d2 xdx− ε(1 − x2 )+x = 0dt2
dt
In this study, we investigate the effect to three star-coupled oscillators by adding another oscillator with different frequency. In this work, we confirm the case of four oscillators and the case of five oscillators in the system. First, we change
the parameter α of the fourth oscillator and investigate the effect to the star-coupled circuits with three oscillators. Next, weadd the fifth oscillator and investigate the effect to the star-coupled oscillators. We consider that many unknown phenomena
remain in such systems. Therefore, it is very important to investigate such systems.

References[1] K. Urabe, Y. Nishio, “Comparing Two-Layer CNN with van der Pol Oscillators Coupled by Inductors,” Proceedings ofInternational Symposium on Nonlinear Theory and its Applications (NOLTA’06), pp. 723-726, Sep. 2006.
[2] R. S. Barbosa, J.A. T. Machado, B.M. Vinagre, “Initial conditions-independent limit cycles of a fractional-order van der
Pol oscillator,” Journal of Vibration and Control, May 2016, 22: 2135-2146, first published on July 15, 2015.
[3] Y. Xu, R. Gu, H. Zhang, W. Xu, J. Duan, “Stochastic bifurcations in a bistable Duffing−van der Pol oscillator withcolored noise,” Phys. Rev. E 83, 056215, Published 26 May 2011.
[4] T.M. Hai, K. Oi, Y. Uwate, Y. Nishio, “Synchronization Phenomena in Star-Couple Van der Pol Oscillators by Adding
Different Frequency Oscillators,” IEEE Asia Pacific Conference on circuits and systems (APCCAS’16), Paper ID 2189,
Oct. 2016.

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