J. Numer. Math., Vol. 22, No. 4, pp. 343–362 (2014) DOI 10.1515 / jnma-2014-0015 c  de Gruyter 2014 On the piecewise-spectral homotopy analysis method and its convergence: solution of hyperchaotic L ¨ u system S. S. MOTSA ∗ , H. SABERI NIK †‡ , S. EFFATI † , and J. SABERI-NADJAFI † Received October 19, 2012 Received in revised form August 13, 2013 Abstract — In this paper, a novel modification of the spectral-homotopy analysis method (SHAM) technique for solving highly nonlinear initial value problems that model systems with chaotic and hyper-chaotic behaviour is presented. The proposed method is based on implementing the SHAM on a sequence of multiple intervals thereby increasing it’s radius of convergence to yield highly accurate method which is referred to as the piece-wise spectral homotopy analysis method (PSHAM). We investigate the application of the PSHAM to the L¨ u system [20] which is well known to display periodic, chaotic and hyper-chaotic behaviour under carefully selected values of it’s governing parameters. Existence and uniqueness of solution of SHAM that give a guarantee of convergence of SHAM, has been discussed in details. Comparisons are made between PSHAM generated results and results from literature and Runge–Kutta generated results and good agreement is observed. Keywords: hyperchaotic system, Banach’s fixed point theorem, piecewise-spectral homo- topy analysis method, spectral collocation 1. Introduction The study of initial value problems (IVPs) that model chaotic motion con- tinues to be an active area of research. Chaos theory studies the behaviour ∗ School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa † Department of Applied Mathematics, School of Mathematical Sciences Ferdowsi Uni- versity of Mashhad, Mashhad, Iran ‡ Corresponding author: saberi hssn@yahoo.com Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 11/29/14 4:31 PM