Existence of Crossing Periodic Solutions of a Duffing Equation with Discontinuity

Classical periodic solution theories of undamped Duffing equations are mostly restricted to continuous nonlinearities, while discontinuous models from mechanical impact and switching circuits lack systematic existence criteria for crossing periodic orbits. This work addresses a second-order undamped Duffing system with jump discontinuity of g(x) on x=0, where g(x) is piecewise continuously differentiable on two half real axes. With the separation condition ruling out sliding trajectories on the discontinuity line, the continuity of the Poincaré map is proved by Filippov theory for discontinuous ODEs. Using the Poincaré–Bohl fixed-point theorem, we derive multiple sufficient conditions ensuring the existence of 2π crossing periodic solutions via successive relaxation of growth hypotheses, and a numerical test example confirms the practicability of our theoretical findings, extending classical continuous Duffing results to discontinuous dynamical systems.