# Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics

## TL;DR

This paper derives explicit analytical expressions for the trajectories, action-angle variables, and chaotic dynamics of the Morse oscillator, enhancing understanding of its behavior under perturbations.

## Contribution

It provides new explicit formulas for Morse oscillator trajectories and establishes conditions for chaos using a Melnikov approach.

## Key findings

- Explicit formulas for trajectories and periods
- Conditions for chaotic dynamics under perturbation
- Application of Melnikov method to Morse oscillator

## Abstract

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach.