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Periodic solutions of autonomous $S^1$-symmetric Newtonian systems

A. Golebiewska, S. Rybicki, P. Stefaniak

TL;DR

The paper studies global bifurcation for continua of nonstationary $2\pi$-periodic solutions of autonomous $S^1$-symmetric Newtonian systems by developing an equivariant degree theory on the torus $T^2$ and its Euler ring $U(T^2)$. A variational formulation yields a $T^2$-invariant functional $\Phi$ whose critical points correspond to periodic orbits, and a $T^2$-equivariant bifurcation index $\mathcal{BIF}_{T^2}$ is constructed to detect continua emanating from stationary states with $ abla U'(u_0)=0$. The main result shows that if $\lambda_0\in\Lambda(u_0)$ and the bifurcation index is nontrivial, then there exists a closed connected continuum of nontrivial solutions either noncompact or intersecting the stationary set, providing a robust existence mechanism even when the Brouwer index vanishes. The paper also discusses the distinction between equivariant and Brouwer indices and demonstrates noncompact continua via explicit potentials, highlighting the practical impact of the equivariant framework for symmetric Hamiltonian-like systems.

Abstract

The aim of this paper is to formulate necessary conditions and sufficient ones for the existence of closed connected sets of nonstationary $2 π$-periodic solutions of $S^1$-symmetric Newtonian systems in $C_{2 π}([0,2π],Ω) \times (0,+ \infty)$. As the main topological tool we apply the degree for equivariant gradient maps.

Periodic solutions of autonomous $S^1$-symmetric Newtonian systems

TL;DR

The paper studies global bifurcation for continua of nonstationary -periodic solutions of autonomous -symmetric Newtonian systems by developing an equivariant degree theory on the torus and its Euler ring . A variational formulation yields a -invariant functional whose critical points correspond to periodic orbits, and a -equivariant bifurcation index is constructed to detect continua emanating from stationary states with . The main result shows that if and the bifurcation index is nontrivial, then there exists a closed connected continuum of nontrivial solutions either noncompact or intersecting the stationary set, providing a robust existence mechanism even when the Brouwer index vanishes. The paper also discusses the distinction between equivariant and Brouwer indices and demonstrates noncompact continua via explicit potentials, highlighting the practical impact of the equivariant framework for symmetric Hamiltonian-like systems.

Abstract

The aim of this paper is to formulate necessary conditions and sufficient ones for the existence of closed connected sets of nonstationary -periodic solutions of -symmetric Newtonian systems in . As the main topological tool we apply the degree for equivariant gradient maps.
Paper Structure (15 sections, 16 theorems, 101 equations)

This paper contains 15 sections, 16 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Notation and elements of equivariant theory
  4. $S^1$-Representations
  5. $T^2$-Representations
  6. The Euler ring $U(T^2)$
  7. The degree for equivariant gradient maps
  8. Periodic solutions of Newtonian systems
  9. Formulation of the main result
  10. Variational setting
  11. Global bifurcation
  12. Proof of Theorem \ref{['thm:main']}
  13. Discussion of the results
  14. Equivariant versus Brouwer index
  15. Existence of noncompact continua

Key Result

Corollary 2.3

Let $\mathbb{V}=(\mathbb{R}^N, \rho_{\mathbb{V}})$ be an orthogonal representation of $S^1$. Then there exist $r \in \mathbb{N}\cup \{0\}$ and finite sequences $(k_0, \ldots, k_r),(m_1, \ldots, m_r),$ where $k_1, \ldots, k_r, m_1, \ldots, m_r \in \mathbb{N}, k_0 \in \mathbb{N} \cup \{0\}, m_1<\ldots

Theorems & Definitions (40)

  • Remark 2.1
  • Corollary 2.3
  • Corollary 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Remark 2.9
  • Lemma 2.10
  • proof
  • ...and 30 more