2014/07/07 by Samuel A. Burden, Shankar Sastry, S. Shankar Sastry +6
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · Mathematics · #93 #Advanced Differential Equations and Dynamical Systems #Control and Dynamics of Mobile Robots #Dynamical Systems (math.DS) #FOS: Mathematics #Microtubule and mitosis dynamics #Nonlinear Dynamics and Pattern Formation #math.DS #msc:93
paper · pdf · doi:10.48550/arxiv.1407.1775
openalex publication_date 2014/07/07 · arxiv created 2015/04/22 · arxiv updated 2015/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a class of discontinuous vector fields brought to our attention by multi-legged animal locomotion. Such vector fields arise not only in biomechanics, but also in robotics, neuroscience, and electrical engineering, to name a few domains of application. Under the conditions that (i) the vector field's discontinuities are locally confined to a finite number of smooth submanifolds and (ii) the vector field is transverse to these surfaces in an appropriate sense, we show that the vector field yields a well-defined flow that is Lipschitz continuous and piecewise-differentiable. This implies that although the flow is not classically differentiable, nevertheless it admits a first-order approximation (known as a Bouligand derivative) that is piecewise-linear and continuous at every point. We exploit this first-order approximation to infer existence of piecewise-differentiable impact maps (including Poincaré maps for periodic orbits), show the flow is locally conjugate (via a piecewise-differentiable homeomorphism) to a flowbox, and assess the effect of perturbations (both infinitesimal and non-infinitesimal) on the flow. We use these results to give a sufficient condition for the exponential stability of a periodic orbit passing through a point of multiply intersecting events, and apply the theory in illustrative examples to demonstrate synchronization in abstract first- and second-order phase oscillator models.