A new method for homoclinic solutions of ordinary differential equations

Abstract

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to file homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit. totally analytic approximation of Blasius' viscous flow problems. Int J Non-Linear Mech 1999;34(4):759-78: Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004:147(2):499-513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006:360:109-13; Hayat T, Sajid M. On analytic solution for thin film flow of a Forth grade fluid down it vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving bell. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function call be obtained as it special case of this homotopy analysis method. Finally, homoclinic solutions arc obtained (for nontrivial examples) analytically by HAM, and arc presented through graphs. (C) 2009 Published by Elsevier Ltd.