Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities

Based on temporal rescaling and harmonic balance, an extended asymptotic perturbation method for parametrically excited two-degree-of-freedom systems with square and cubic nonlinearities is proposed to study the nonlinear dynamics under 1:2 internal resonance. This asymptotic perturbation method is employed to transform the two-degree-of-freedom nonlinear systems into a four-dimensional nonlinear averaged equation governing the amplitudes and phases of the approximation solutions. Linear stable analysis at equilibrium solutions of the averaged equation is done to show bifurcations of periodic motion and homoclinic motions. Furthermore, analytical expressions of homoclinic orbits and heteroclinic cycles for the averaged equation without dampings are obtained. Considering the action of the damping, the bifurcations of limit cycles are also investigated. A concrete example is further provided to discuss the correctness and accuracy of the extended asymptotic perturbation method in the case of small-amplitude motion for the two-degree-of-freedom nonlinear system.