Af. El-bassiouny et Hm. Abdelhafez, Prediction of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic and quartic non-linearities, MATH COMP S, 57(1-2), 2001, pp. 61-80
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.