Resonance simulation of the coupled nonlinear Mathieu’s equation

Abstract

Numerous theoretical physics and chemistry problems can be modeled using Mathieu’s equations (MEs). They are crucial to the theory of potential energy in quantum systems, which is equivalent to the Schrödinger equation. According to the mentioned applications, thus, the current study investigates the stability behavior of the nonlinear-coupled MEs. The analysis of the coupled harmonic resonance cases imposes two coupled solvability conditions, which leads to coupled parametric nonlinear Landau equations. In addition, a super-harmonic nonlinear resonance combination is presented. Solutions and stability criteria are discussed for each case. It is shown that resonance produces an unstable system. The transition curves are derived. Numerical calculations show the excitation of the frequency on the periodic solutions.