Fast-slow dynamics with non-zero sliding characteristics related to pitchfork bifurcation delay in the parametrically excited Duffing system with frequency switching

Pitchfork bifurcation delay, resulted from the slow passage through a supercritical pitchfork bifurcation point, often leads to fast-slow oscillations with zero-type sliding motions, e.g., the classic pitchfork-bifurcation-delay-induced bursting. This paper aims to report fast-slow dynamics with non-zero sliding motions related to the pitchfork bifurcation delay by taking the slowly parametrically excited Duffing system with frequency switching as an example. Typically, the coexistence of two symmetric pitchfork-bifurcation-delay-induced bursting can be observed in the slowly parametrically excited Duffing system. We show that the symmetric solution structure related to the pitchfork bifurcation may be destroyed in the presence of frequency switching. As a result, pitchfork-bifurcation-delay-induced switchings to different equilibrium branches may lead to different bursting patterns. In particular, interesting non-zero sliding motions, characterized by that the associated system trajectory slides along the frequency-switching threshold for some time, can be observed in the bursting. We reveal dynamical mechanisms of the non-zero sliding motions by theories of fast-slow systems and of piecewise-smooth systems, and explore their transitions by considering the frequency-switching threshold as the control parameter. Our study shows that different frequency-switching thresholds may lead the system trajectory to switch to different solution branches, e.g., the equilibrium branches and the switching boundary. As a result, three different bursting patterns with non-zero sliding motions are obtained.