The stability of boundary equilibria of three-dimensional Filippov systems

For three-dimensional piecewise-smooth systems of ordinary differential equations, this paper characterises the stability of points that belong to a switching surface and are equilibria of exactly one of the two neighbouring pieces of the system. Stability is challenging to characterise when nearby orbits repeatedly switch between regular motion on one side of the switching surface, and sliding motion on the switching surface, as defined via Filippov’s convention. We prove that in this case stability is governed by the behaviour of a global reinjection mechanism of a four-parameter family of piecewise-linear hybrid systems, and perform a detailed numerical study of this family.

💡 Research Summary

This paper addresses a subtle stability problem that arises in three‑dimensional Filippov systems when an equilibrium lies on a switching surface but is an equilibrium of only one of the two smooth vector fields that define the system. Such points, called boundary equilibria, occur at so‑called boundary equilibrium bifurcations, where an equilibrium of one side collides with the discontinuity manifold as a parameter varies. The difficulty stems from the fact that nearby trajectories may repeatedly alternate between regular flow on one side of the surface and sliding motion on the surface itself, according to Filippov’s convex‑combination rule. Classical linearisation (eigenvalues of the Jacobian) is insufficient because the dynamics involve a non‑trivial interaction between the regular and sliding regimes.

The authors first set up the general Filippov system

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