Prediction of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic and quartic non-linearities

Two methods (the multiple scales and the generalized synchronization) are used to investigate second-order approximate analytic solution. The first-order ordinary differential equations are derived for evaluation of the amplitude and phase with damping, non-linearity and all possible resonances. These equations are used to obtain stationary solution. The results obtained by these two methods are in excellent agreement. The instability regions of the response of the considered oscillator are determined via an algorithm that use Floquet theory to evaluate the stability of the investigated second-order approximate analytic solutions in the neighborhood of the non-linear resonance of the system. Bifurcation diagram is constructed for one-degree-of-freedom system with quadratic, cubic and Quartic non-linearities under the interaction of external and parametric excitations. The numerical solution of the system is obtained applying Runge–Kutta method carries the predictions, which exhibit chaos motions among other behavior. Graphical representations of the results are presented.