Contraction and Periodic Orbits in Time-Periodic Filippov Systems

Abstract

We develop an explicit and verifiable contraction framework for planar, time-periodic Filippov systems with a single codimension-one switching manifold. On compact forward-invariant sets, the framework treats smooth flow, switching, and sliding within a single calculus by combining (i) differential contraction off the manifold via Clarke’s generalized Jacobian, (ii) multiplicative jump contraction at switching instants encoded directly in a weighted distance, and (iii) tangential contraction along the Filippov sliding vector field. The analysis is saltation-free and yields computable exponential rates directly from model data.
Three structural results underpin the theory. (1) Local-to-global contraction: a local uniform exponential bound extends to global contraction on compact convex invariant sets. (2) Metric transfer: contraction in the weighted distance implies exponential decay in the Euclidean norm with the same rate up to fixed constants, ensuring physical interpretability. (3) Average contraction under bounded switching: if smooth/sliding segments contract at rate ν > 0, each switch contributes at most a factor eκ, and the switch count satisfies N(t) ≤ ρt+N0, then trajectories decay at the effective rate νeff = ν-ρκ. Under T-periodic forcing, the stroboscopic map is a contraction, and Banach’s theorem yields a unique exponentially orbitally stable T- periodic Filippov solution with an explicit convergence rate.
The assumptions (time periodicity, piecewise-C1 regularity, a single switching manifold, compact forward invariance, and a mild dwell-time/no-Zeno condition) are minimal for our purposes. An explicit two-dimensional piecewise-smooth oscillator (mass–spring–damper with Coulomb friction) demonstrates closed-form verification of the hypotheses; simulations visualize contraction across smooth and sliding regimes.
MSC: 34A36, 34D20, 37C60.