Homoclinic bifurcation in a linearly damped Morse oscillator driven by an amplitude modulated force

Abstract

We investigate homoclinic bifurcations in a Morse oscillator subjected to an amplitude-modulated force, using both analytical and numerical approaches. The system features a degenerate fixed point at infinity, which is regularized via a McGehee-type transformation, thus rendering the Melnikov method applicable. By employing the Melnikov analytical technique, we derive the threshold condition for the onset of horseshoe chaos. Melnikov threshold curves are plotted across various external parameter spaces. The analytical predictions are validated through direct numerical simulations. Parametric domains where horseshoe chaos is mitigated are identified. Additionally, the system reveals rich dynamical behavior, including period-doubling and intermittency routes to chaos, along with the emergence of periodic windows under the influence of the amplitude-modulated force. Numerical results, consisting of phase portraits, Poincare maps, and bifurcation diagrams, substantiate the analytical findings and disclose intricate dynamical phenomena such as period-doubling cascades, interspersed chaotic regions, and the presence of homoclinic bifurcations.