Heteroclinic Orbits for a Discrete Pendulum Equation

TL;DR

This paper proves the existence of heteroclinic orbits in a discrete pendulum equation using variational methods, extending classical results from continuous systems to discrete analogs.

Contribution

It demonstrates the existence of heteroclinic orbits in the discrete pendulum equation, a novel extension from continuous to discrete systems using variational techniques.

Findings

Heteroclinic orbits exist between specific points in the discrete pendulum system.

Variational methods effectively establish the existence of these orbits.

Results extend classical continuous system findings to discrete models.

Abstract

About twenty years ago, Rabinowitz showed firstly that there exist heteroclinic orbits of autonomous Hamiltonian system joining two equilibria. A special case of autonomous Hamiltonian system is the classical pendulum equation. The phase plane analysis of pendulum equation shows the existence of heteroclinic orbits joining two equilibria, which coincide with the result of Rabinowitz. However, the phase plane of discrete pendulum equation is similar to that of the classical pendulum equation, which suggests the existence of heteroclinic orbits for discrete pendulum equation also. By using variational method and delicate analysis technique, we show that there indeed exist heteroclinic orbits of discrete pendulum equation joining every two adjacent points of $\{2k\pi+\pi: k\in{\mathbb Z}\}$.