  The interplay of positive and negative feedback loops on different time
scales appears to be a fundamental mechanisms for robust and tunable
oscillations in both biological systems and electro-mechanical systems. We
develop a detailed analysis of a basic three dimensional Lure model to show how
controlled oscillations arise from the tuning of positive and negative feedback
strengths. Our analysis is based on dominance theory and confirms, from a
system-theoretic perspective, that the mixed feedback is a fundamental enabler
of robust oscillations. Our results are not limited to three dimensional
systems and extend to larger systems via passivity theory, and to uncertain
systems via small gain arguments.
