Bifurcation Solutions of Non-linear Fourth Order Ordinary Differential Equation with Two Parameters

Year 2018, , 113 - 120, 27.04.2018

Mudhir A. Abdul Hussain

https://doi.org/10.36753/mathenot.421779

Abstract

In this article we are interested in the study of bifurcation solutions of non-linear fourth order differential
equation by using local method of Lyapunov -Schmidt. The reduced equation corresponding to the
non-linear fourth order differential equation is given by a nonlinear system of two algebraic equations.
The classification of the solutions (equilibrium points) of this system has been discussed.

Keywords

Bifurcation theory, Nonlinear systems, Local Lyapunov-Schmidt method

References

• [1] B. S. Bardin and S. D. Furta, Local existence theory for periodic wave moving of an infinite beam on a nonlinearly elastic support, Actual Problems of Classical and Celestial Mechanics, Elf, Moscow, 13-22 (1998).
• [2] B.M. Darinskii, C.L. Tcarev, Yu.I. Sapronov, Bifurcations of Extremals of Fredholm Functionals,Journal of Mathematical Sciences, Vol. 145, No. 6, 2007.
• [3] B.V. Loginov, Theory of Branching Nonlinear Equations in Theconditions of Invariance Group, - Tashkent: Fan, 1985.
• [4] J. M. T. Thompson and H. B. Stewart , Nonlinear Dynamics and Chaos, Chichester, Singapore, J. Wiley and Sons, 1986. V.R. M.A. Abdul Hussain, Corner Singularities of Smooth Functions in the Analysis of Bifurcations Balance of the Elastic Beams and Periodic Waves, Ph. D. thesis, Voronezh- Russia. 2005.
• [5] M.A. Abdul Hussain,Bifurcation Solutions of Boundary Value Problem, Journal of Vestnik Voronezh, Voronezh State University, No. 1, 2007, 162-166, Russia.
• [6] M.A. Abdul Hussain, Bifurcation Solutions of Elastic Beams Equation with Small Perturbation, Int. J. Math. Anal. (Ruse) 3 (18) (2009), 879-888.
• [7] M.A. Abdul Hussain, Two Modes Bifurcation Solutions of Elastic Beams Equation with Nonlinear Approximation, Communications in Mathematics and Applications journal, Vol. 1, no. 2, 2010, 123-131. India.
• [8] M.A. Abdul Hussain, Two-Mode Bifurcation in Solution of a Perturbed Nonlinear Fourth Order Differential Equation, Archivum Mathematicum (BRNO), Tomus 48 (2012), 27-37, Czech Republic.
• [9] M.A. Abdul Hussain, A Method for Finding Nonlinear Approximation of Bifurcation Solutions of Some Nonlinear Differential Equations, Journal of Applied Mathematics and Bioinformatics, vol.3, no.3, 2013, UK.
• [10] M.A. Abdul Hussain, Nonlinear Ritz Approximation for Fredholm Functionals, Electronic Journal of Differential Equations, Vol. 2015, No. 294,USA,2015, 1-11.
• [11] M.M. Vainberg, V.A. Trenogin, Theory of Branching Solutions of Nonlinear Equations, M.-Science , 1969.
• [12] Yu.I. Sapronov, Regular Perturbation of Fredholm Maps and Theorem about odd Field, Works Dept. of Math., Voronezh Univ., 1973. V. 10, 82-88.
• [13] Yu.I. Sapronov, Finite Dimensional Reduction in the Smooth Extremely Problems, - Uspehi math., Science, - 1996, V. 51, No. 1., 101- 132.
• [14] Yu.I. Sapronov and V.R. Zachepa , Local Analysis of Fredholm Equation, Voronezh Univ., 2002.