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 u sed to investigate second-order approximate analytic solution. The first-or der ordinary differential equations are derived for evaluation of the ampli tude and phase with damping, non-linearity and all possible resonances. The se equations are used to obtain stationary solution. The results obtained b y these two methods are in excellent agreement. The instability regions of the response of the considered oscillator are determined via an algorithm t hat 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 solut ion of the system is obtained applying Runge-Kutta method carries the predi ctions, which exhibit chaos motions among other behavior. Graphical represe ntations of the results are presented. (C) 2001 IMACS. Published by Elsevie r Science B.V All rights reserved.