Non-trivial solutions and their stability in a two-degree-of-freedom Mathieu–Duffing system

The Mathieu–Duffing equation represents a basic form for a parametrically excited system with cubic nonlinearities. In multi-degree-of-freedom systems, parametric resonances and the associated limit cycles take place at both principal and combination resonance frequencies. Furthermore, using asynchronous parametric excitation of coupling terms leads to a broadband destabilization of the trivial solution and the appearance of limit cycles at non-resonant frequencies. Regarding applications, the utilization of this excitation method has its significant importance in micro- and nanosystems. On the one hand, cubic nonlinearities are found to be abundant in these systems. On the other hand, parametric excitation is preferably utilized in these systems for better amplification leading to an enhanced sensitivity and for squeezing thermal noise, and thus, proved to be significantly useful in mechanical, optical and microwave systems. Therefore, this theoretical investigation should be of relevant importance to those small-scaled systems. Accordingly, a general two-degree-of-freedom Mathieu–Duffing system is studied. The non-trivial solutions are obtained at different parametric resonance conditions. A bifurcation analysis is carried out using the multiple scales method, followed by investigating the effect of the asynchronous parametric excitation on the existence of limit cycles at resonant and non-resonant frequencies.