On the stability of hybrid polycycles

TL;DR

The paper addresses stability characterization of polycycles in planar hybrid dynamical systems that combine smooth flow and jump dynamics, allowing hyperbolic saddles, tangential, and jump singularities. It introduces a generalized graphic number $r=\prod r_i$, where each $r_i$ is a hyperbolicity ratio that accounts for saddles and jump singularities, and proves a universal stability criterion: the polycycle is stable if $r>1$ and unstable if $r<1$, with infinity-power jumps contributing stability. The proof analyzes the first return map via Dulac maps near saddles and power-law transition maps near jump singularities, and specializes to known frameworks such as Filippov and smooth systems. The results extend the understanding of limit cycles and graphics in hybrid dynamics and are illustrated with a bouncing-ball example and a figure-eight polycycle, highlighting potential applications.

Abstract

In this paper we provide the stability of generic polycycles of hybrid planar vector fields, extending previous known results in the literature. The polycycles considered here may have hyperbolic saddles, tangential singularities and jump singularities.

Figures (17)

• Figure 1: Illustration of a graph given by the connection of the separatrix of a saddle-node with its parabolic sector.
• Figure 2: Illustration of a polycycle $(a)$ homeomorphic and $(b)$ not homeomorphic to $\mathbb{S}^1$.
• Figure 3: Illustration of a polycycle on planar Filippov systems.
• Figure 4: Illustration of an orbit of a hybrid system. The solid lines represent the orbit given by the flow of a smooth vector field. The dashed lines represent the instantaneous jump of a point $p$, given by a map $\varphi_i$, when it reaches the switching manifold $\Sigma=\Sigma_1\cup\Sigma_2$.
• Figure 5: Illustration of polycycles in hybrid systems. Here $\overline{p_i}$ denotes the image of $p_i$ after the jump. Observe that a jump may keep $p_i$ fixed while reversing the orientation, see $p_2$ on the left-hand side. In particular, polycycles can now be one-sided.

Theorems & Definitions (12)

• Example 1: Example $1$ of LliSan2024
• Definition 1
• Remark 1
• Definition 2
• Definition 3: Definition of Polycycle
• Definition 4
• Definition 5
• Theorem 1
• Proposition 1: Lemma $2.3$, Section $2.2$ of Soto
• proof : Proof of Theorem \ref{['T1']}