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On the periodic orbits of C0-typical impulsive semiflows

Jaqueline Siqueira, Maria Joana Torres, Paulo Varandas

TL;DR

This work investigates impulsive semiflows formed by a Lipschitz flow on a compact manifold together with a (continuous) impulsive map, focusing on how typical (i.e., $C^0$-generic) perturbations influence the density of periodic orbits. The authors develop a Poincaré-map framework tailored to impulsive dynamics and prove two parallel generic-density results: (i) for fixed Lipschitz flows, a $C^0$-generic impulse yields dense periodic orbits on the impulsive non-wandering set within the impulsive region; (ii) for fixed impulses, a $C^0$-generic Lipschitz flow yields dense periodic orbits in the non-wandering set. Central to the proofs are $C^0$-perturbation lemmas—one for impulses (closing lemma) and one for vector fields (flowbox-based perturbations)—and a permanence notion ensuring robustness of periodic points under perturbations. The results unify and extend previous $C^1$-type findings by showing similar density phenomena under $C^0$ perturbations, with substantial differences between perturbing impulses and perturbing vector fields due to their distinct impact on the dynamics. The paper also provides concrete examples (e.g., impulsive Anosov flows and minimal flows) and discusses potential extensions to volume-preserving settings and shadowing properties, highlighting avenues for further research in impulsive topological dynamics.

Abstract

Impulsive semiflows modeled by a Lipschitz continuous vector field and continuous impulse functions, defined over an impulsive region, are piece-wise Lipschitz continuous semiflows with piecewise smooth trajectories. In this paper we contribute to the topological description of typical impulsive semi-flows, parameterized by both vector fields and impulses. We prove that $C^0$-generic Lipschitz vector fields generate impulsive semiflows with denseness of periodic orbits on the non-wandering set. Additionally, we show that $C^0$-generic impulses generate impulsive semiflows with denseness of periodic orbits on the impulsive non-wandering set.

On the periodic orbits of C0-typical impulsive semiflows

TL;DR

This work investigates impulsive semiflows formed by a Lipschitz flow on a compact manifold together with a (continuous) impulsive map, focusing on how typical (i.e., -generic) perturbations influence the density of periodic orbits. The authors develop a Poincaré-map framework tailored to impulsive dynamics and prove two parallel generic-density results: (i) for fixed Lipschitz flows, a -generic impulse yields dense periodic orbits on the impulsive non-wandering set within the impulsive region; (ii) for fixed impulses, a -generic Lipschitz flow yields dense periodic orbits in the non-wandering set. Central to the proofs are -perturbation lemmas—one for impulses (closing lemma) and one for vector fields (flowbox-based perturbations)—and a permanence notion ensuring robustness of periodic points under perturbations. The results unify and extend previous -type findings by showing similar density phenomena under perturbations, with substantial differences between perturbing impulses and perturbing vector fields due to their distinct impact on the dynamics. The paper also provides concrete examples (e.g., impulsive Anosov flows and minimal flows) and discusses potential extensions to volume-preserving settings and shadowing properties, highlighting avenues for further research in impulsive topological dynamics.

Abstract

Impulsive semiflows modeled by a Lipschitz continuous vector field and continuous impulse functions, defined over an impulsive region, are piece-wise Lipschitz continuous semiflows with piecewise smooth trajectories. In this paper we contribute to the topological description of typical impulsive semi-flows, parameterized by both vector fields and impulses. We prove that -generic Lipschitz vector fields generate impulsive semiflows with denseness of periodic orbits on the non-wandering set. Additionally, we show that -generic impulses generate impulsive semiflows with denseness of periodic orbits on the impulsive non-wandering set.
Paper Structure (16 sections, 11 theorems, 33 equations, 3 figures)

This paper contains 16 sections, 11 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Impulsive semiflows
  4. Space of impulsive maps
  5. Space of Lipschitz continuous vector fields
  6. Statement of the main results
  7. $C^0$-Baire generic impulses
  8. $C^0$-Baire generic Lipschitz flows
  9. Examples
  10. Preliminares
  11. Poincaré maps for impulsive semiflows
  12. $C^0$-perturbations lemmas
  13. Permanent periodic orbits
  14. $C^0$-generic impulses
  15. $C^0$-generic Lipschitz flows
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\varphi$ be a Lipschitz continuous flow generated by $X\in \mathfrak{X}^{0,1}(M)$ and $D,\hat{D}\subset M$ be smooth codimension one submanifolds transversal to the flow such that hypothesis eq:noncompact holds. There exists a $C^0$-Baire generic subset $\mathscr R_X \subset \mathscr I_{D,\hat for every $I \in \mathscr R_X$, where ${Per(\psi_{X,I})}$ denotes the set of periodic orbits of $\p

Figures (3)

  • Figure 1: Representation of the impulsive flow on the section $\{z=0\}$, where $D_0=D\cap \{z=0\}$ and $\hat{D}_0=\hat{D}\cap \{z=0\}$. Existence of non-permanent periodic orbits in $\Omega(\psi_{X,I}) \cap \hat{D}_0$.
  • Figure 2: Three types of periodic orbits for the impulsive semiflows
  • Figure 3: Flowbox on the last lap of the periodic orbit

Theorems & Definitions (24)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 4.1
  • proof
  • Lemma 4.1
  • proof
  • ...and 14 more