Bias of the Steady-State Averaged Solutions of a Strongly Overdamped Particle in a Cosine Potential Under Harmonic Excitation

The existence and stability of steady-state averaged solutions for a strongly overdamped particle in a cosine potential are analytically studied when subjected to harmonic excitation. The stability loss of these solutions, which oscillate near the bottom of the potential well, can be viewed as an escape problem. The harmonic balance method is used for the analysis up to the first order, including a constant bias term.
When a massless particle is assumed, the steady-state solutions oscillate around either the potential well's minima or maxima. However, taking the particle's mass into account alters the structure of the steady-state solution. Within certain excitation parameter ranges, a new set of equilibrium positions for the oscillation center arises, linking the equilibria at the potential's extremities. Interestingly, modulating only the force amplitude can continuously bias the steady-state oscillation center.