Cyclicity of sliding cycles with singularities of regularized piecewise smooth visible-invisible two-folds

TL;DR

This work analyzes the cyclicity of sliding cycles near visible-invisible two-folds in regularized piecewise-smooth systems by transforming the regularized dynamics into a slow-fast problem via cylindrical blow-up. A central tool is the connection between the divergence integral along slow-fast orbits and transition maps, which reduces the problem to counting zeros of a one-dimensional function derived from a pair of divergence integrals. By classifying sliding cycles according to the location and multiplicity of singularities of the sliding vector field, the authors establish explicit upper bounds on the number of limit cycles (cyclicity) in multiple geometric configurations, and demonstrate applicability to regularized PWL two-folds. The results extend the canard-like analysis of sliding cycles to settings with singular slow dynamics away from the two-fold, providing a unified framework for bounding cyclicity via slow-divergence data and transition-map structure. This has potential impact for understanding limit cycles in regularized PWS models and informing the design of systems where controlled switching behavior is essential.

Abstract

In this paper we study the cyclicity of sliding cycles for regularized piecewise smooth visible-invisible two-folds, in the presence of singularities of the Filippov sliding vector field located away from two-folds. We obtain a slow-fast system after cylindrical blow-up and use a well-known connection between the divergence integral along orbits and transition maps for vector fields. Since properties of the divergence integral depend on the location and multiplicity of singularities, we divide the sliding cycles into different classes, which can then produce different types of cyclicity results. As an example, we apply our results to regularized piecewise linear systems.

Figures (6)

• Figure 1.1: Two-fold singularities. From the left to the right: visible-visible, visible-invisible and invisible-invisible. Each case has sub cases that depend on the direction of the flow of $Z_{\lambda,c}^\pm$ and $X_{\lambda,c}^{sl}$.
• Figure 1.2: (a) A PWS system with a visible-invisible two-fold singularity at $(x,y)=(0,0)$, and the sliding cycle $\Gamma$. Sliding vector field $X_{0,c_0}^{sl}$ has two singularities in $[\eta_{-},\eta_{+}]$: one corner singularity $x=\eta_-$ and one singularity in the interior $(\eta_{-},\eta_{+})$. (b) Canard cycle $\widetilde{\Gamma}$ in a smooth slow-fast system. The slow dynamics contains singularities.
• Figure 2.1: Different types of sliding cycles $\Gamma$, depending on the location of singularities of the sliding vector field $X_{0,c_0}^{sl}$. The red line is the switching manifold $y=0$ and blue curves are orbits of $Z^{-}_{0,c_0}$. Red dots are two-folds, black dots represent singularities of $X_{0,c_0}^{sl}$.
• Figure 3.1: Cylindrical blow-up along $(\Sigma_{sl}\times \{0\})\cup\{(0,0,0)\}$ for the extended system \ref{['extended']}. We use the case III of Figure \ref{['FIG3']} as an example, where $\eta_{-}$ is a singularity of $X_{0,c_0}^{sl}(x)$ with odd multiplicity.
• Figure 4.1: Cases in Theorem \ref{['thm-appl']}. Observe that in cases I, II, IV, VI, VII and IX the singularity $x^{*}$ is positioned at $\eta_{-}$, however the case in which $x^{*}$ is positioned at $\eta_{+}$ is also allowed. Similarly, In cases III, V, VII and X the singularity $x^{*}$ is positioned at $\eta_{+}$, but it could be positioned at $\eta_{-}$ as well.

Theorems & Definitions (24)

• Definition 1
• Theorem 2.1
• Remark 1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 3.1
• proof
• Remark 2