Second-order Filippov systems: sliding dynamics without sliding regions

TL;DR

This paper develops a mathematical framework for second-order Filippov systems, describing their unique sliding dynamics without sliding regions, and applies the theory to mechanical and biological models.

Contribution

It introduces a new theory for second-order Filippov systems with continuous Lie derivatives, characterizes their dynamics, and connects to sliding mode control.

Findings

Crossing orbits spiral around tangency surfaces.

No Zeno behavior; orbits do not converge in finite time.

Derived a vector field for second-order sliding motion.

Abstract

This paper develops fundamental mathematical theory for second-order Filippov systems. These are discontinuous ordinary differential equations with solutions defined in the sense of Filippov, and whose first Lie derivatives vary continuously across discontinuity surfaces. Unlike generic Filippov systems, discontinuity surfaces consist only of crossing regions and their boundaries where both adjacent vector fields are tangent to the discontinuity surface. Crossing orbits spiral around invisible-invisible tangency surfaces, and we derive a formula for the attractive or repulsive strength of these surfaces. We prove crossing orbits cannot converge to tangency surfaces in finite time (no Zeno), and that the limiting dynamics consists of Filippov solutions on the tangency surfaces (second-order sliding motion). We derive a vector field that governs this motion, and characterise the stability of equilibria on tangency surfaces. The methodology is applied to a model of a mechanical oscillator with compliant impacts, and a model of ant colony migration. We also relate second-order Filippov systems to second-order sliding mode control, and show that for two-dimensional systems the results reduce to known theory.