International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 12, Pages 739-761

Parametrically excited nonlinear systems: a comparison of two methods

A. F. El-Bassiouny1,2

1Mathematics Department, Faculty of Science, Benha University, Benha 13518, Egypt
2Department of Mathematics, Jubail College of Education for Girls, P.O. Box 12020, Jubail Industrial city 31961, Saudi Arabia

Received 9 March 2001

Copyright © 2002 A. F. El-Bassiouny. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Subharmonic resonance of two-degree-of-freedom systems with cubic nonlinearities to multifrequency parametric excitations in the presence of three-to-one internal resonance is investigated. Two approximate methods (the multiple scales and the generalized synchronization) are used to construct a first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases. Steady state solutions and their stability are computed for selected values of the system parameters. The results obtained by the two methods are in excellent agreement. Numerical solutions are carried out and graphical representations of the results are presented and discussed.