The hybrid matching of Hurwitz systems

TL;DR

This work studies planar hybrid systems formed by two Hurwitz linear vector fields connected by a jump, providing a complete analytic criterion for global asymptotic stability and revealing limit-cycle bifurcations. By embedding planar piecewise vector fields into a hybrid structure, the authors reduce the dynamics to a global normal form and classify Hurwitz vector fields into types N_1, N_2, and F, deriving explicit behavior in each regime. The central result shows that if at least one side is not type F, the origin is GAS; if both are type F, the system exhibits GAS, global instability, or a global center, and, crucially, a unique hyperbolic limit cycle whose existence and nature are governed by a displacement map and the parameters r=1/s and ρ. The analysis enables precise analytic criteria for stability and bifurcation, and demonstrates that the hybrid formulation can produce limit cycles that are impossible in the purely piecewise framework, with potential implications for control and qualitative dynamics.

Abstract

In this paper we study planar hybrid systems composed by two stable linear systems, defined by Hurwitz matrices, in addition with a jump that can be a piecewise linear, a polynomial or an analytic function. We provide an explicit analytic necessary and sufficient condition for this class of hybrid systems to be asymptotically stable. We also prove the existence of limit cycles in this class of hybrid systems. Our results can be seen as generalizations of results already obtained in the literature. This was possible due to an embedding of piecewise smooth vector fields in a hybrid structure.

Figures (7)

• Figure 1: Illustration of an orbit of the bouncing ball model. For simplicity, we interchanged the coordinate axes.
• Figure 2: Illustration of $\varphi_\rho$ and $\mathfrak{X}$. The origin $\mathcal{O}$ is interpreted as a singularity. By abuse of notation we drew two copies of $\Sigma_\rho$.
• Figure 3: Illustration of two Hurwitz vector fields $X^\pm$ of type $N_1$. The curves in red (resp. blue) are the solutions of $X^+$ (resp. $X^-$). For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.
• Figure 4: Illustration of two Hurwitz vector fields $X^\pm$ of type $N_2$. The curves in red (resp. blue) are the solutions of $X^+$ (resp. $X^-$). For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.
• Figure 5: Illustration of two Hurwitz vector fields $X^\pm$ of type $F$. The curves in red (resp. blue) are the solutions of $X^+$ (resp. $X^-$). For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.

Theorems & Definitions (12)

• Conjecture 1: Markus-Yamabe MarYam1960
• Example 1: Example $1$ of LliSan2025
• Theorem A
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4
• proof : Proof of Theorem \ref{['Main1']}
• Remark 1: The case $\rho=0$