An instrumental insight for a periodic solution of a fractal Mathieu-Duffing equation

The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu-Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu-Duffing equation in the continuous space by employing a new modification of He's definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, "the non-perturbative approach", that deals with the obtained equation has been introduced.