Non-resonant damped transitions resembling continuous resonance scattering in coupled oscillators with essential nonlinearities

Abstract

We study a peculiar damped nonlinear transition of a system of two coupled oscillators into a state of sustained nonlinear resonance scattering. This system consists of a grounded, weakly damped linear oscillator attached to a light, weakly damped oscillator with essential (nonlinearizable) stiffness nonlinearity of the third degree, and linear or nonlinear damping. We find that under specific forcing conditions the damped response of this system locks into a damped, non-resonant transition resembling continuous resonance scattering, whereby the transient damped dynamics closely follows an impulsive orbit manifold of the dynamics in the frequency–energy plane. This manifold is formed by a countable infinity of periodic orbits and an uncountable infinity of quasi-periodic orbits of the underlying Hamiltonian system, with each of these orbits representing the response of the Hamiltonian system being initially at rest and forced by an impulse applied to the linear oscillator. Hence, the damped transitions reported here appear to lock in sustained resonance scattering from a countable infinity of periodic orbits along the impulsive orbit manifold. Such transitions represent an anti-resonance state, where the dynamics is farthest away from resonance. We conjecture that such transitions are only made possible by the essential (nonlinearizable) stiffness nonlinearity of the nonlinear attachment and cannot be realized in linearizable nonlinear dynamics where resonance captures prevent sustained resonance scattering. Our findings are supported by numerical, analytical and experimental results.