Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local bifurcations in a class of piecewise-smooth Filippov systems with a nonregular switching curve via a nonlinear double regularization process

Claudio A. Buzzi, Yagor Romano Carvalho

TL;DR

The paper investigates local bifurcations in planar piecewise-smooth Filippov systems with a nonregular switching curve under a nonlinear double regularization that smooths the dynamics. It derives equilibrium criteria for the regularized vector field and analyzes how the codimension of low-order bifurcations can be preserved or lost depending on the subclass of the Filippov field and on the transition functions. Key contributions include a detailed criterion for the existence of equilibria in the regularized system, a characterization of when structural stability is preserved via the Flow Box Theorem, and explicit examples including Hopf-type and saddle-node-type bifurcations. The findings highlight the nonuniform impact of smoothing on codimension, with implications for modeling and interpreting regularized piecewise-smooth systems in engineering and physics.

Abstract

We are interested in analyzing the preservation of bifurcations in a class of piecewise smooth vector fields with a nonregular switching set under a smoothing process that approximates them by smooth vector fields. We examine cases in which the codimension is either preserved or altered, as well as whether the generic nature of the bifurcation is maintained.

Local bifurcations in a class of piecewise-smooth Filippov systems with a nonregular switching curve via a nonlinear double regularization process

TL;DR

The paper investigates local bifurcations in planar piecewise-smooth Filippov systems with a nonregular switching curve under a nonlinear double regularization that smooths the dynamics. It derives equilibrium criteria for the regularized vector field and analyzes how the codimension of low-order bifurcations can be preserved or lost depending on the subclass of the Filippov field and on the transition functions. Key contributions include a detailed criterion for the existence of equilibria in the regularized system, a characterization of when structural stability is preserved via the Flow Box Theorem, and explicit examples including Hopf-type and saddle-node-type bifurcations. The findings highlight the nonuniform impact of smoothing on codimension, with implications for modeling and interpreting regularized piecewise-smooth systems in engineering and physics.

Abstract

We are interested in analyzing the preservation of bifurcations in a class of piecewise smooth vector fields with a nonregular switching set under a smoothing process that approximates them by smooth vector fields. We examine cases in which the codimension is either preserved or altered, as well as whether the generic nature of the bifurcation is maintained.
Paper Structure (5 sections, 8 theorems, 92 equations, 14 figures, 4 tables)

This paper contains 5 sections, 8 theorems, 92 equations, 14 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Filippov Systems, Local Structural Stability, and Codimension-One Bifurcations
  3. Proofs of Theorems \ref{['teonep']}, \ref{['teo1']}, \ref{['teo2']} and \ref{['teo4']}.
  4. Aspects of an equilibrium point at the origin
  5. Examples of regularization

Key Result

Theorem 1.1

Consider $Z \in \Omega$ as in sistemacruz. Then, its nonlinear double regularization $Z^R_{\varepsilon, \eta}$ has an equilibrium point if and only if there exists $p_0 = (p_{10}, p_{20}) \in \mathcal{U}$ such that $\det[Z](p_0) = 0$ and $H_i(p_0) = 0$ for some $i = 1,2$. Moreover, if $\xi \neq 0$,

Figures (14)

  • Figure 1: Decomposition of the set $\mathcal{U}$.
  • Figure 2: Sotomayor--Teixeira double regularization.
  • Figure 3: (a)-(b) Monotonous transition functions; (c) Nonmonotonous transition function; (d) Generalized transition function .
  • Figure 4: Examples of Filippov conventions and orbits at origin.
  • Figure 5: The origin is a fold point of both $X$ and $Y$ in cases (a) and (b). In case (c), the origin is a regular-fold of $X$.
  • ...and 9 more figures

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 10 more