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Closed stable orbits in a strongly coupled resonant Wilberforce pendulum

Misael Avendaño-Camacho, Alejandra Torres-Manotas, José A Vallejo

Abstract

We prove the existence of closed stable orbits in a strongly coupled Wilberforce pendulum, for the case of a $1:2$ resonance, by using techniques of geometric singular symplectic reduction combined with the more classical averaging method of Moser.

Closed stable orbits in a strongly coupled resonant Wilberforce pendulum

Abstract

We prove the existence of closed stable orbits in a strongly coupled Wilberforce pendulum, for the case of a resonance, by using techniques of geometric singular symplectic reduction combined with the more classical averaging method of Moser.

Paper Structure

This paper contains 8 sections, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Numerical analysis
  3. Normal forms in perturbation theory
  4. Invariants of the Hamiltonian flow of the harmonic oscillator with two degrees of freedom.
  5. Second-order normal form of the Hamiltonian
  6. Constructing the reduced phase space
  7. Stability analysis at the degenerate points
  8. Stability analysis at the singular point

Figures (4)

  • Figure 1: Strongly coupled Wilberforce pendulum dynamics for different values of the parameter $\varepsilon$. The resonance 1:2 is shown.
  • Figure 2: Poincaré maps of the strongly coupled Wilberforce pendulum for different values of $\varepsilon$. The resonance $1:2$ is shown.
  • Figure 3: Reduced phase space of the 1:2 resonance.
  • Figure 4: Critical points of the first-order normal Hamiltonian $N_1$.