International Journal of Scientific & Technology Research

IJSTR@Facebook IJSTR@Twitter IJSTR@Linkedin
Home About Us Scope Editorial Board Blog/Latest News Contact Us

IJSTR >> Volume 3- Issue 10, October 2014 Edition

International Journal of Scientific & Technology Research  
International Journal of Scientific & Technology Research

Website: http://www.ijstr.org

ISSN 2277-8616

Construction Of Lyapunov Functions For Some Fourth Order Nonlinear Ordinary Differential Equations By Method Of Integration

[Full Text]



Orie Bassey O.



Index Terms: Lyapunov Functions, Linear systems, non linear systems Negative definite, Positive definite, Linear and non linear systems



Abstract: In giving adequate attention to some qualitative properties of solutions in ordinary differential equations, Lyapunov functions is quite indispensable. The process of tackling some problems in the application is the construction of appropriate Lyapunov functions for some nonlinear fourth order differential equation. Our focus is finding V(x), a quadratic form and positive definite also finding U(x) which is positive definite such that the derivative of V with respect to time would be equal to the negative value of U(x) In this work, we adopted the pre-multiplication of the given differential equation by and thereafter we integrated with respect to form to . We obtained a Lyapunov function candidate for a fourth order differential equation or its scalar equation.



[1] Ademola, A.T., Aramowo, P.O. “Asymptotic Behavior of Solutions of Third Order Non-Linear Differential Equation” Acta Univ. Sapientiae, Mathematica, 3,2, (2011), 197-211.

[2] Afuwape, A.U, Omeike, M.O, “Stability and Boundedness of Solutions of a Kind of Third Order Delay Differential Equations. Computational & Applied Mathematics, Vol. 29, N. 3 pp 329-342, 2010

[3] Bainov, D.D “Second Method of Lyapunov and Existence of Periodic Solutions of Linear Impulsive Differential – Difference Equations”. Divulgaciones Mathematicas. V.S, No. ½ (1997), 29-36.
[4] Grimm, V. Quispel, G.R.W. “Geometric Integration Methods that Preserve Lyapunov Functions, BIT Numerical Mathematics, 2005 Vol. 45, No. 1, pp 709-723.

[5] Hedrick, J.K, Girarad A. “Stability of Non Linear Sys-tems” Control of Non linear Dynamic Systems. Theory and Applications, 2005.

[6] Jingl, Ren, Wing-Sum Cheung, Zhibo Cheng “Ex-istence and Lyapunov stability of periodic solutions for Generalized higher-order Neutral Differential equations” Hindawi Publishing Corporation. Boun-dary value problems, volume 2011 Article ID635767, 21 pages, doi: 10: 1155 2011/635767

[7] Ezeolu J.O.C, Ogbu, H.M “Construction of Lyapunov – Type of Functions for some third order Non linear ordinary differential equations by the method of integration”, Journal of Science Teachers Association of Nigeria 45, 1 & 2, April & September 2010, 49-58.

[8] Lars Grune, Peter, E. Kloedem, Stefan Siegmund, Fabian R. Wirth. “Lyapunov’s Second method for Non autonomous Differential Equations” Communicated by Aim Sciences Support by the Science Foundation’s Ireland grants 04-IN3-1460 and 00/PI.1/6067, 2006.

[9] Mohammed, Dahleh, Munthier A. Dahlel, George Verghese “Lectures on Dynamic Systems and con-trol” Department of Electrical Engineering and Computer Science. Massachusetts Institute of Technology.

[10] Mohammed, Harmouche, Salah Laghrouche, Yacine Chitour “Stabilization of Perturbed Integrator Chains using Lyapunov Based Homogeneous Controllers, ar Xi v: 1303.5330VZ (Math O.C) 30th May 2013.

[11] Ogundare, B.S “On Boundaries and Stability of Solutions of Certain Third Order Delay Differential Equation “Journal of the Nigerian Mathematical Society, Vol. 31 pp 55-68, 2012.

[12] Omeike, M.O “Further Results on Global Stability of Solutions of Certain Third Order Non Linear Diffe-rential Equations”. Acta Univ. Palacki Olomuc, Fac, rer, nat. Mathematica 47 (2008) 121-127.