Chaos and fundamental bifurcation phenomena from a relaxation oscillator with periodic thresholds

A relaxation oscillator which includes a hysteresis element with periodic variation of the threshold is analyzed. Deriving a one-dimensional return map, the parameter regions are clarified where the system exhibits a periodic solution or quasi-periodic solutions and where the response characteristics of the system having periodic solutions becomes devil's stair and where the attractors may coexist. When the system has chaotic response, then coexistence of chaotic and periodic attractors is confirmed. Various results are confirmed from the laboratory experiments.