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Codimension Two Bifurcation Problems in Forced Nonlinear Circuits
Tetsuya YOSHINAGA Hiroshi KAWAKAMI
Publication
IEICE TRANSACTIONS (19761990)
Vol.E73
No.6
pp.817824 Publication Date: 1990/06/25
Online ISSN:
DOI:
Print ISSN: 00000000 Type of Manuscript: Special Section PAPER (Special Issue on Engineering Chaos) Category: Chaos in Electrical Circuits Keyword:
Full Text: PDF(589.7KB)>>
Summary:
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 3dimensional differential equation. For 3dimensional system, however, two types of bifurcations never occur. In this paper, we shall treat 4dimensional 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.

