Research Article Received 18 July 2011 Published online 20 April 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.1581 MOS subject classification: 34A34; 34B15; 34C28; 32L81; 65L60 A numerical scheme for solutions of the Chen system ¸ Suayip Yüzba¸ sı * † Communicated by J. Cash In this paper, we will develop the Bessel collocation method to find approximate solutions of the Chen system, which is a three-dimensional system of ODEs with quadratic nonlinearities. This scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. By help of the collocation points and the matrix operations of derivatives, the unknown coeffi- cients of the Bessel polynomials are calculated. The accuracy and efficiency of the proposed approach are demonstrated by two numerical examples and performed with the aid of a computer code written in MAPLE. In addition, compar- isons between our method and the homotopy perturbation method numerical solutions are made with the accuracy of solutions. Copyright © 2012 John Wiley & Sons, Ltd. Keywords: Chen system; Bessel collocation method; approximate solution; nonlinear differential equation systems; Bessel polynomials and series; collocation points 1. Introduction Many physical and engineering events can be modelled by chaotic or nonchaotic systems of ODEs. In this paper, we consider the Chen dynamical system, first found by Chen et al. [1–3], defined as 8 ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ : dx dt D a.y  x/ dy dt D .c  a/x  xz C cy dz dt D xy  bz, , x.0/ D  1 , y.0/ D  2 , z.0/ D  3 , 0  t  R, (1) where x, y and z are state variables, a, b, c and R are positive constants,  1 ,  2 and  3 are appropriate constants. Recently, Noorani and coworkers studied the Adomian decomposition method [4–6] for the solutions of the Chen system. Chowdhury and Hashim [7] have solved the Chen system using the multistage homotopy perturbation method. Additionally, some authors have worked on chaotic dynamical systems in [8–11]. Also, Yüzba¸ si et al. [12–18] have studied the Bessel collocation method to solve the linear neutral delay differential equations, singular differential-difference equations, pantograph equations, Lan–Emden differential equations, linear differential equation systems, linear Volterra integral equation systems and linear Fredholm integro-differential equation systems. In this paper, we will find the appproximate solutions of system (1) by developing the Bessel collocation method given in [12–18]. Our purpose is to find approximate solutions of system (1) expressed in the truncated Bessel series form x.t/ D N X nD0 a 1,n J n .t/, y.t/ D N X nD0 a 2,n J n .t/ and z.t/ D N X nD0 a 3,n J n .t/ (2) Department of Mathematics, Faculty of Science, Mu˘ gla University, Mu˘ gla, Turkey *Correspondence to: ¸ Suayip Yüzba¸ sı, Department of Mathematics, Faculty of Science, Mu˘ gla University, Mu˘ gla, Turkey. † E-mail: suayip@mu.edu.tr Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012, 35 885–893 885