Sliding fast–slow dynamics in the slowly forced Duffing system with frequency switching.

This paper aims to report sliding fast–slow dynamics related to the classic relaxation oscillations. To this end, the slowly forced Duffing system with frequency switching is taken as an example. Typical fast–slow oscillations, i.e., the classic relaxation oscillations related to an S-shaped equilibrium hysteresis curve with two fold points, can be observed in the slowly forced Duffing system. We show that, in the presence of frequency switching, the fast–slow oscillations may exhibit distinct sliding behaviors, characterized by that the oscillation trajectory slides along the frequency-switching threshold for some time. Then, the frequency-switching threshold is taken as the control parameter. As a result, six different patterns of sliding fast–slow oscillations are obtained, and their dynamical mechanisms are revealed by theories of fast–slow dynamical systems and of piecewise-smooth dynamical systems. Our study shows that the frequency-switching threshold has great effects on dynamical characteristics of the switching boundary. In particular, the switching boundary can be divided into several areas whose dynamical characteristics may vary along with the frequency-switching threshold. Besides, dynamical behaviors of subsystems near the switching boundary depend heavily on the frequency-switching threshold. These factors account for the generation of different sliding fast–slow oscillation patterns in relation to the variation of frequency-switching threshold. [ABSTRACT FROM AUTHOR]