approximate solutions of coupled nonlinear oscillations: ‎stability analysis

The current article is concerned with a comprehensive investigation in achieving approximate solutions of coupled nonlinear oscillations with high nonlinearity. these equations are highly nonlinear secondorder ordinary differential equations. via a coupling of the homotopy perturbation method and laplace transforms, which is socalled the helaplace method, traditional approximate solutions involving the secular terms are accomplished. on the other hand, in order to cancel the secular terms, an expanded frequency technique is adapted to accomplish periodic approximate solutions. therefore, a nonlinear frequency, for each differential equation, is achieved. furthermore, for more convenience, these solutions are pictured to indicate their behavior. the multiple timescales with the aid of the homotopy concept are utilized to judge the stability criteria. the analyses reveal the resonance as well as the nonresonant cases. additionally, numerical calculations are carried out, graphically, to address the regions that guaranteed the bounded solutions. it is found that the latter method, is the most powerful mathematical tool in extracting the stability analysis of the considered system.