Codimension Two Bifurcation Problems in Forced Nonlinear Circuits


IEICE TRANSACTIONS (1976-1990)   Vol.E73   No.6   pp.817-824
Publication Date: 1990/06/25
Online ISSN: 
Print ISSN: 0000-0000
Type of Manuscript: Special Section PAPER (Special Issue on Engineering Chaos)
Category: Chaos in Electrical Circuits

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In a nonlinear dynamical circuit with sinusoidal external source, we frequently encounter various bifurcation phenomena of steady states such as jump and hysteresis phenomenon, frequency entrainment, etc. The steady state corresponds to a periodic solution of the circuit equations described by nonlinear ordinary differential equations. The generic bifurcations of the periodic solution are known as codimension one bifurcations: tangent bifurcation, period doubling bifurcation and the Hopf bifurcation. At a bifurcation value of parameters, if a periodic solution satisfies two bifurcation conditions, then the bifurcation refers as a codimension two bifurcation. This type of bifurcation may be observed in high dimensional systems with several parameters. In Ref.(1), we have classified codimension two bifurcations and proposed a numerical method for obtaining the bifurcation parameters. To illustrate the occurrences of some types of codimension two bifurcations, we analyzed a circuit described by 3-dimensional differential equation. For 3-dimensional system, however, two types of bifurcations never occur. In this paper, we shall treat 4-dimensional system as an illustrating example. In this example, we shall see all types of codimension two bifurcations defined in this paper. For a global property of bifurcation set of parameters, it is found that a type of codimension two bifurcation occurs successively together with the period doubling cascade and the Hopf bifurcations. This bifurcation sequence may cause a new route to the generation of chaotic oscillations.