Codimension Two Bifurcation Problems in Forced Nonlinear Circuits

Tetsuya YOSHINAGA; Hiroshi KAWAKAMI

Codimension Two Bifurcation Problems in Forced Nonlinear Circuits

Tetsuya YOSHINAGA, Hiroshi KAWAKAMI

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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.

Publication: IEICE TRANSACTIONS on transactions Vol.E73-E No.6 pp.817-824

Publication Date: 1990/06/25

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Type of Manuscript: Special Section PAPER (Special Issue on Engineering Chaos)

Category: Chaos in Electrical Circuits

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Tetsuya YOSHINAGA
Hiroshi KAWAKAMI

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Tetsuya YOSHINAGA, Hiroshi KAWAKAMI, "Codimension Two Bifurcation Problems in Forced Nonlinear Circuits" in IEICE TRANSACTIONS on transactions, vol. E73-E, no. 6, pp. 817-824, June 1990, doi: .
Abstract: 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.
URL: https://globals.ieice.org/en_transactions/transactions/10.1587/e73-e_6_817/_p

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@ARTICLE{e73-e_6_817,
author={Tetsuya YOSHINAGA, Hiroshi KAWAKAMI, },
journal={IEICE TRANSACTIONS on transactions},
title={Codimension Two Bifurcation Problems in Forced Nonlinear Circuits},
year={1990},
volume={E73-E},
number={6},
pages={817-824},
abstract={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.},
keywords={},
doi={},
ISSN={},
month={June},}

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TY - JOUR
TI - Codimension Two Bifurcation Problems in Forced Nonlinear Circuits
T2 - IEICE TRANSACTIONS on transactions
SP - 817
EP - 824
AU - Tetsuya YOSHINAGA
AU - Hiroshi KAWAKAMI
PY - 1990
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E73-E
IS - 6
JA - IEICE TRANSACTIONS on transactions
Y1 - June 1990
AB - 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.
ER -