Global bifurcations of a periodically forced nonlinear oscillator
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.
two-dimensional ordinary differential equation
scientific article; zbMATH DE number 3916067
scientific article