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Analysis of bifurcations in oscillatory circuits

Hiroyuki Asahara, Takuji Kousaka, Tetsushi Ueta

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Abstract

In this chapter, we investigate the bifurcation phenomena observed in oscillatory circuits. The stability and bifurcation phenomena in autonomous systems are introduced by focusing on the equilibrium point and the fixed point. The characteristics and conditions of the saddle-node bifurcation, Hopf bifurcation, and pitchfork bifurcation are discussed for the equilibrium point. Likewise, the characteristics and conditions of the saddle-node bifurcation, period-doubling bifurcation, Neimark-Sacker bifurcation, and pitchfork bifurcation are introduced for the fixed point. The method for computing the bifurcation points of the equilibrium point and the periodic points is also introduced, and an example of an application is presented.

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What this paper is about

In this chapter, we investigate the bifurcation phenomena observed in oscillatory circuits. The stability and bifurcation phenomena in autonomous systems are introduced by focusing on the equilibrium point and the fixed point. The characteristics and conditions of the saddle-node bifurcation, Hopf bifurcation, and pitchfork bifurcation are discussed for the equilibrium point. Likewise, the characteristics and conditions of the saddle-node bifurcation, period-doubling bifurcation, Neimark-Sacker bifurcation, and pitchfork bifurcation are introduced for the fixed point. The method for computing the bifurcation points of the equilibrium point and the periodic points is also introduced, and an example of an application is presented.

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Available abstract

In this chapter, we investigate the bifurcation phenomena observed in oscillatory circuits. The stability and bifurcation phenomena in autonomous systems are introduced by focusing on the equilibrium point and the fixed point. The characteristics and conditions of the saddle-node bifurcation, Hopf bifurcation, and pitchfork bifurcation are discussed for the equilibrium point. Likewise, the characteristics and conditions of the saddle-node bifurcation, period-doubling bifurcation, Neimark-Sacker bifurcation, and pitchfork bifurcation are introduced for the fixed point. The method for computing the bifurcation points of the equilibrium point and the periodic points is also introduced, and an example of an application is presented.

Key concepts: Pitchfork bifurcation, Saddle-node bifurcation, Transcritical bifurcation, Bogdanov–Takens bifurcation, Biological applications of bifurcation theory, Bifurcation diagram, Period-doubling bifurcation, Infinite-period bifurcation

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