Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method Fathi M. Allan Department of Mathematical Sciences, UAE University, Al-Ain, P.O. Box 17551, United Arab Emirates Accepted 18 June 2007 Abstract Based on a new kind of analytic method, namely the Homotopy analysis method, an analytic approach to solve non- linear, chaotic system of ordinary differential equations is presented. The method is applied to Lorenz system; this sys- tem depends on the three parameters: r, b and the so-called bifurcation parameter R are real constants. Two cases are considered. The first case is when R = 20.5 which corresponds to the transition region and the second case corresponds to R = 23.5 which corresponds to the chaotic region. The validity of the method is verified by comparing the approximation series solution with the results obtained using the standard numerical techniques such as Runge-Kutta method. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Most of the scientific problems and phenomena are modeled by non-linear ordinary or partial differential equations. In most cases, these problems do not admit analytical solution, so these non-linear equations should be solved using special techniques. In recent years, much attention has been devoted to the newly developed methods to construct an analytic solutions of non-linear equation, such methods include the Adomian decomposition method [1–3,5,23– 29], the Homotopy analysis method (HAM), [4a,4b,15–17,22,30,31], the Homotopy perturbation method (HPM) [10,11], and the variational iteration method (VIM) [7–9,12–14]. Perturbation techniques are too strongly dependent upon the so-called ‘‘small parameters’’ [19]. Thus, it is worth- while developing some new analytic techniques independent upon small parameters. Liao [15–17] proposed such a kind of analytic technique, namely the Homotopy analysis method (HAM) which is valid for more of non-linear problems, especially those with strong non-linearity. Furthermore, the HAM provides us with great freedom to select related ini- tial approximations, governing equations of auxiliary sub-problems and also some auxiliary parameters. It is just this kind of freedom which provides us with a larger possibility to ensure the corresponding approximation sequence of the HAM convergent. The validity of the Homotopy analysis method was tested by many authors [4,16,17,30,31]. The HAM yields, without linearization, perturbation, transformation or discretization, an analytical solution in terms of an infinite power series with easily computable terms. 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.116 E-mail address: f.allan@uaeu.ac.ae Chaos, Solitons and Fractals 39 (2009) 1744–1752 www.elsevier.com/locate/chaos