Global bifurcations of a periodically forced nonlinear oscillator
type: Item
instance of: scholarly article
title: Global bifurcations of a periodically forced nonlinear oscillator
zbMATH Open document ID: 0573.92005
DOI: 10.1007/BF00277669
author: James P. Keener
author: Leon Glass
published in: Journal of Mathematical Biology
publication date: 1984-01-01T00:00:00Z
review text: The authors investigate the effects of periodic stimulation on the so called radial isochron clock (RIC) in considerable detail. This system is a prototypical example for models of biological oscillations, which is described by a two-dimensional ordinary differential equation, whose only limit sets are an isolated unstable singular point and an isolated stable limit cycle. It has already been shown by \textit{M. R. Guevara} and the second author [ibid. 14, 1-23 (1982; Zbl 0489.92007)] and by \textit{F. C. Hoppensteadt} and the first author [ibid. 15, 339-349 (1982; Zbl 0489.92006)] that periodic stimulation on the RIC leads to period doubling bifurcations and chaotic dynamics. In the paper under review the structure of the global bifurcations in the parameter space and in the neighborhood of the transition from type 1 (i.e., the phase transition curve, PTC, has topological degree 1) to type 0 (PTC has degree 0) are investigated by numerical and analytical methods. It is shown that the bifurcations are generic for a class of models which generalize the RIC.
Mathematics Subject Classification ID: 92B05
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 3916067
zbMATH Keywords: phase-locking
zbMATH Keywords: chaos
zbMATH Keywords: periodic stimulation
zbMATH Keywords: radial isochron clock
zbMATH Keywords: biological oscillations
zbMATH Keywords: two-dimensional ordinary differential equation
zbMATH Keywords: limit sets
zbMATH Keywords: isolated unstable singular point
zbMATH Keywords: isolated stable limit cycle
zbMATH Keywords: RIC
zbMATH Keywords: period doubling bifurcations
zbMATH Keywords: global bifurcations
zbMATH Keywords: phase transition curve
zbMATH Keywords: topological degree
reviewed by: Reinhard Bürger
Wikidata QID: Q52698009
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00277669
OpenAlex ID: W2085165830
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description: scientific article; zbMATH DE number 3916067