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

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 birhythmicity, phase reversals, radial trapping, phase jumps and spiraling patterns in the amplitude space. Our results may find useful applications in physical, chemical or biological systems.

VERSION 4: Fig. 10(d) added, an uncited reference removed; (To appear in Physica D) (17 pages, 21 figures, two column, aps RevTeX); VERSION 3: Revised. In Section 2, small tau approximation added; Section 3 is divided into subsections; periodic solution discussed in detail; Figs. 7 and 11 discarded; Figs. 12 and 14 altered; three new figures (now Figs. 10, 11 and 21) added. VERSION 2: Figs. 1 and 2 replaced

References in corpus (4)

• Time Delay in the Kuramoto Model of Coupled Oscillators
• Time Delay Induced Death in Coupled Limit Cycle Oscillators
• Time Delay Effects on Coupled Limit Cycle Oscillators at Hopf Bifurcation
• Phase slips and phase synchronization of coupled oscillators

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