On the dynamics of systems with vastly different frequencies

Abstract

The method of direct partition of motion (DPM) has been widely used to study the dynamics of non-autonomous oscillatory systems subject to external high frequency excitation. In this work, we explore the non-trivial dynamics that arise in autonomous systems with vastly different frequencies, in which the fast excitation is intrinsic to the system and possibly influenced by the slow dynamics. Using three model problems, we illustrate how DPM could be useful for the analysis of such systems and when combined with WKB can serve to capture the strong modulation of a fast oscillator due to coupling to a much slower one -a phenomenon that is not readily handled using standard perturbation methods. First, we study a system of three coupled limit cycle oscillators, which when uncoupled, have the frequencies [omega]1 = O (1), [omega]2 = O(1/[epsilon]) and [omega]3 = O(1/[epsilon]2), respectively, where [epsilon] << 1. Approximate expressions for the limit cycles of oscillators 1 and 2 are found in terms of Jacobi elliptic functions. For coupling strengths exceeding critical bifurcation values, the limit cycle of oscillator 1 or 2 is found to disappear. In the second problem, we consider a simple pendulum coupled to a horizontal mass-spring system of a much higher frequency. In contrast to the first problem, the coupling here allows the slow oscillator to affect the leading order dynamics of the fast oscillator. This calls for a rescaling of fast time, inspired by the WKB method, to be employed in conjunction with DPM. We obtain a critical energy value at which a pitchfork bifurcation of periodic orbits is found to occur, giving rise to non-local periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and [pi]/2 from the downwards position. Finally, the developed method is utilized to explain the non-trivial dynamics arising in a model of a thin elastica presented by Cusumano and Moon in 1995. We observe that for the corresponding experimental system, the ratio of the two natural frequencies of the system was [ALMOST EQUAL TO] 44 which can be considered to be of O (1/[epsilon]) where [epsilon] << 1. Hence, the system is best viewed as one with vastly different frequencies. The method leads to an approximate expression for the non-local modes of the type observed in the experiments as well as the bifurcation energy value at which these modes are born. The formal approximate solutions obtained for these problems are validated by comparison with numerical integration.