Dynamics of a Hysteretic Relay Oscillator with Periodic Forcing

The dynamics of a hysteretic relay oscillator with simple harmonic forcing is studied in this paper. Even though there are no bounded solutions in the absence of forcing, periodic excitation gives rise to more complex responses including periodic, quasi-periodic, and chaotic behavior. A Poincaré map is introduced to facilitate mathematical analysis. Families of period-one solutions are determined as fixed points of the Poincaré map. These represent coexisting subharmonic responses. Conditions on the amplitude and frequency of the forcing for the existence of periodic solutions have been obtained. Linear stability analysis reveals that these solutions can be classified as centers or saddles. The presence of higher periodic or quasi-periodic motions together with homoclinic and heteroclinic tangles implies the existence of chaotic solutions.