Melnikov Function Analysis of Chaos in a Parametrically Excited Morse Oscillator with Two Forcing Terms

Abstract

This paper focuses on the analysis of chaos using the Melnikov method in a linearly damped Morse oscillator subjected to two parametric forcing terms. 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. By using the Melnikov analytical technique, we derive the threshold condition for the onset of horseshoe chaos. We analyze the effects of two parametric forces with frequencies Ω = ω and Ω = ω on the dynamics of the system. Due to the two parametric forces, we identify regions where chaos is either suppressed or enhanced, depending on the parametric choices. The analysis is carried out in terms of bifurcation diagrams, phase portraits, trajectory plots, and the time measure 1/τM. The results indicate that the system with two parametric forces exhibits rich dynamical behaviors, including multiple coexisting attractors, period-doubling bifurcations, period-bubbling, reverse perioddoubling bifurcations, antimonotonicity (i.e., concurrent creation and annihilation of periodic orbits), and chaotic attractors.