Exponential stability of nonsmooth periodic orbits in the complex space

We study the nonsmooth, time-periodic differential equation z ˙ ( t ) = sin ⁡ ( ω t ) − m sign ( Re ( z ( t ) ) ) − η i Im ( z ( t ) ) , z ( t ) ∈ C , motivated by dry-friction dynamics and defined on C ≅ R 2 . The system exhibits a codimension-one discontinuity, leading to regimes of smooth flow, sliding, and transversal crossing in the Filippov sense. Despite the lack of scalar ordering in higher dimensions, we establish global exponential stability of a unique periodic orbit using two complementary approaches: spectral analysis of the Poincaré map and a locally defined Riemannian contraction metric. These methods extend Giesl's scalar framework [7] to nonsmooth systems in R 2 . • Limitations of Giesl's scalar contraction framework are shown for nonsmooth systems in C ≅ R 2 . • A geometric extension to two dimensions is achieved using a time-dependent Riemannian contraction metric. • Exponential stability of a nonsmooth periodic orbit is proved via spectral and metric-based methods.