Dynamics of a Limit Cycle Oscillator under Time Delayed Linear and Nonlinear Feedbacks

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  • 1 August 1999
Abstract
We study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single Hopf bifurcation oscillator. Our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips, frequency suppression, multiple periodic states and chaos. Such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators. Our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator.We construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms. We find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like phase reversals, radial trapping, spiraling patterns in the amplitude space, and birhythmicity. Our results may find useful applications in physical, chemical or biological systems.

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