Analytic approximate solutions of the cubic–quintic Duffing–van​ der Pol equation with two-external periodic forcing terms: Stability analysis

In the light of the potential applications in engineering, electronics, physics, chemistry, and biology, the current work applies several techniques to achieve analytic approximate and numerical solutions of the cubic–quintic Duffing–van der Pol equation. This equation represents a second-order ordinary differential equation with quintic nonlinearity and includes two external periodic forcing terms. A classical approximate solution involves the secular terms is obtained. Unfortunately, this traditional method does not enable us to ignore these secular terms. Additionally, along with the concept of the expanded frequency, a bounded approximate solution is achieved. The Homotopy perturbation method is utilized to obtain an approximate solution with an artificial frequency of the given system. Near the equilibrium points, in the case of the autonomous system, the linearized stability is accomplished. Furthermore, in the case of the non-autonomous system, by means of the multiple time scales, the stability analysis is effectuated, together with the resonance and the non-resonance cases. Numerical computations are performed to demonstrate, graphically, the perturbed solutions as well as the stability/instability regions. Various numerical solutions to initial–boundary value problems are deduced via a three-step finite difference scheme. These are plotted and discussed to show the chaotic nature of solutions.