Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits

We present a contraction-based framework for analyzing periodic behavior in planar nonsmooth dynamical systems governed by Filippov differential inclusions. The approach combines a time- and state-dependent weighted Riemannian metric with Clarke’s generalized Jacobian, together with a uniform jump condition across switching manifolds, to obtain sufficient conditions for exponential contraction on compact forward-invariant sets. These local estimates are assembled into a global stability result via a Banach fixed-point argument, establishing the existence and uniqueness of an exponentially attracting periodic orbit. The theory extends classical contraction results from the scalar setting to two-dimensional systems with discontinuities, sliding motion, and switching phenomena. Beyond clarifying the role of contraction in nonsmooth dynamics, the framework provides a systematic analytic tool for stability verification, with potential applications to hybrid control, nonsmooth mechanical oscillators, and computational methods for piecewise-smooth systems.