  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.
