# Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation

## Ahmed Abbas Mizeal and Mudhir A. Abdul Hussain

Address:

A. A. Mizeal: University of Thi-qar, College of Computer Science and Mathematics, Department of Mathematics, Thi-qar, IRAQ

M. A. A. Hussain: University of Basrah, College of Education, Department of Mathematics, Basrah, IRAQ

E-mail:

aam_mb@yahoo.com

mud_abd@yahoo.com

Abstract: In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.

AMSclassification: primary 34K18; secondary 93C10.

Keywords: bifurcation theory, nonlinear systems, local Lyapunov-Schmidt method.