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Non-resonant Hopf Links Near a Hamiltonian Equilibrium Point

C. Grotta-Ragazzo, Lei Liu, Pedro A. S. Salomão

TL;DR

The paper develops explicit, low-order Birkhoff-Gustavson normal-form criteria that guarantee non-resonant Hopf-linked periodic orbits near a Hamiltonian equilibrium in two degrees of freedom. By analyzing rotation numbers and their twists, and by exploiting weak non-resonance, symmetries, and reductions (including lens-space quotients), it demonstrates the persistence of non-resonant Hopf links under perturbations and extends the results to classical celestial-mechanics models. The main contributions are (i) concrete normal-form conditions ensuring infinitely many periodic orbits on nearby energy surfaces, (ii) a detailed framework for computing rotation-number twists and their impact on resonance, and (iii) application of these methods to the Spatial Isosceles Three-Body Problem, Hill's Lunar Problem, and the Henon-Heiles system, confirming the abundance of periodic orbits in these physically relevant settings.

Abstract

This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface sufficiently close to the equilibrium contains at least two periodic orbits forming a Hopf link (A. Weinstein [19]). A theorem by Hofer, Wysocki, and Zehnder [9] implies that there are either precisely two or infinitely many periodic orbits on such a component of the energy surface. This multiplicity result follows from the existence of a disk-like global surface of section. If a certain non-resonance condition on the rotation numbers of the orbits of the Hopf link is satisfied [8], then infinitely many periodic orbits follow. This paper aims to present explicit conditions on the Birkhoff-Gustavson normal forms of the Hamiltonian function at the equilibrium point that ensure the existence of infinitely many periodic orbits on the energy surface by checking the non-resonance condition as in [8] and not making use of any global surface of section. The main results focus on strongly resonant equilibrium points and apply to the Spatial Isosceles Three-Body Problem, Hill's Lunar Problem, and the Hénon-Heiles System.

Non-resonant Hopf Links Near a Hamiltonian Equilibrium Point

TL;DR

The paper develops explicit, low-order Birkhoff-Gustavson normal-form criteria that guarantee non-resonant Hopf-linked periodic orbits near a Hamiltonian equilibrium in two degrees of freedom. By analyzing rotation numbers and their twists, and by exploiting weak non-resonance, symmetries, and reductions (including lens-space quotients), it demonstrates the persistence of non-resonant Hopf links under perturbations and extends the results to classical celestial-mechanics models. The main contributions are (i) concrete normal-form conditions ensuring infinitely many periodic orbits on nearby energy surfaces, (ii) a detailed framework for computing rotation-number twists and their impact on resonance, and (iii) application of these methods to the Spatial Isosceles Three-Body Problem, Hill's Lunar Problem, and the Henon-Heiles system, confirming the abundance of periodic orbits in these physically relevant settings.

Abstract

This paper is about the existence of periodic orbits near an equilibrium point of a two-degree-of-freedom Hamiltonian system. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface sufficiently close to the equilibrium contains at least two periodic orbits forming a Hopf link (A. Weinstein [19]). A theorem by Hofer, Wysocki, and Zehnder [9] implies that there are either precisely two or infinitely many periodic orbits on such a component of the energy surface. This multiplicity result follows from the existence of a disk-like global surface of section. If a certain non-resonance condition on the rotation numbers of the orbits of the Hopf link is satisfied [8], then infinitely many periodic orbits follow. This paper aims to present explicit conditions on the Birkhoff-Gustavson normal forms of the Hamiltonian function at the equilibrium point that ensure the existence of infinitely many periodic orbits on the energy surface by checking the non-resonance condition as in [8] and not making use of any global surface of section. The main results focus on strongly resonant equilibrium points and apply to the Spatial Isosceles Three-Body Problem, Hill's Lunar Problem, and the Hénon-Heiles System.

Paper Structure

This paper contains 16 sections, 27 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Birkhoff-Gustavson normal forms
  4. The construction of the normal form
  5. Non-resonant Hopf links for Lg
  6. Rotation numbers
  7. The case Lg
  8. The general linearized equation
  9. Non-resonant Hopf links for Lg
  10. Weakly non-resonant Hamiltonians
  11. Hamiltonians with symmetries
  12. Lg-symmetric Hamiltonians
  13. The spatial isosceles three-body problem
  14. Hill's lunar problem
  15. The Heńon-Heiles Hamiltonian
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that $H=H_2+\mathcal{O}_3$ is weakly non-resonant at the origin, i.e., $\alpha_2$ is not a multiple of $\alpha_1$. Assume further that for some $N\geq 3$, the Birkhoff-Gustavson normal form $H= H_{N}+ \mathcal{O}_{N+1}$ satisfies one of the following conditions: Then, for every $E>0$ sufficiently small, the sphere-like component $\Sigma_E \subset H^{-1}(E)$ carries a pair of periodic orbit

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 2.1
  • Theorem 2.2: Birkhoff-Gustavson Normal Form
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • ...and 38 more