S. Siva Sakthi Pitchammal, S. M. Abdul Kader, A. Ponchitra, A. Zeenath Bazeera, V. Chinnathambi
Complex Dynamics in a Linearly Damped Morse Oscillator With Multiple Excitations
Dynamic Systems and Applications 34 (2024) 61-81

https://doi.org/10.46719/dsa2025.34.03

ABSTRACT.
In this paper, we examine the complex dynamics of a linearly damped Morse oscillator subjected under parametric and external excitations. The inclusion of a parametric excitation term adds complexity and interest to the analysis of the Morse oscillator. The unperturbed Morse oscillator features a degenerate fixed point at infinity, which is resolved by applying a McGehee-type
transformation. This transformation regularizes the stationary fixed point, making it possible to apply the Melnikov method effectively to the system. Using analytical techniques, including the Melnikov theory, we derive threshold conditions for the occurrence of horseshoe chaos in the perturbed Morse oscillator. From these threshold conditions, we analyze the onset of horseshoe chaos numerically by measuring the time, τM, between successive changes in the sign of M(τ). Results show that as the depth of the potential well (a) increases, the threshold for horseshoe chaos also increases when varying the amplitude (f) of the external excitation. Conversely, the threshold for horseshoe chaos decreases as the amplitude (η) of the parametric excitation increases. The analytical findings are illustrated through numerical simulations, employing nonlinear analysis tools such as bifurcation diagrams, phase portraits, Poincare maps, and measuring the time τM between successive
sign changes in M(τ).

AMS (MOS) Subject Classification. 34C37, 34D10, 37C29, 37D45, 37J20, 65P20.

Key Words and Phrases. Perturbed Morse oscillator, Horseshoe chaos, Melnikov method, Para-
metric excitation, Chaos