Isochronous sets in smooth and non-smooth mechanical oscillators

Abstract This work aims at characterizing the patterns of iso-phases of periodic and noisy oscillations of smooth and non-smooth dynamical systems in the phase space. The values of asymptotic phases reached starting from various initial conditions form patterns in the basins of attraction in the phase space. Even though the phase of system's oscillations turn out to be functions of the phase of forcing for externally excited systems, for higher periodic oscillations some patterns or foliations within their basins are seen. For periodic oscillations, isophases coincide with the isochrons. For irregular oscillations, a notion of regularity of oscillations has been introduced and the sensitivity of the system to external forcing has been computed. For stable periodic oscillations of autonomous kind, in the presence of noise the extent of phase diffusion is observed to be orders higher than the diffusion in amplitude. For noisy forced oscillations, the phase is locked onto the deterministic phase with marginal dislocations within a limit ε dependent on the intensity of noise. This work deals with the above notions of iso-phases in the context of a Van der Pol and an autonomous stick-slip belt friction oscillator.