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Research Papers

Continuous Galerkin Petrov Time Discretization Scheme for the Solutions of the Chen System

[+] Author and Article Information
S. Hussain

Department of Mathematics,
Mohammad Ali Jinnah University,
Islamabad 44000, Pakistan
e-mail: shafqat.hussain@jinnah.edu.pk

Z. Salleh

School of Informatics and Applied Mathematics,
University Malaysia Terengganu,
Kuala Terengganu 21030,
Terengganu, Malaysia
e-mail: zabidin@umt.edu.my

Manuscript received May 13, 2014; final manuscript received February 1, 2015; published online April 9, 2015. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 10(6), 061007 (Nov 01, 2015) (7 pages) Paper No: CND-14-1123; doi: 10.1115/1.4029714 History: Received May 13, 2014; Revised February 01, 2015; Online April 09, 2015

In this paper, the continuous Galerkin Petrov time discretization (cGP) scheme is applied to the Chen system, which is a three-dimensional system of ordinary differential equations (ODEs) with quadratic nonlinearities. In particular, we implement and analyze numerically the higher order cGP(2)-method which is found to be of fourth order at the discrete time points. A numerical comparison with classical fourth-order Runge–Kutta (RK4) is given for the presented problem. We look at the accuracy of the cGP(2) as the Chen system changes from a nonchaotic system to a chaotic one. It is shown that the cGP(2) method gains accurate results at larger time step sizes for both cases.

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Figures

Grahic Jump Location
Fig. 1

A nonchaotic case with parameters a = 35, b = 12, and c = 28. (Left) error solutions using cGP(2) (h = 0.005) and RK4 (h = 0.0005) and (right) error solutions using cGP(2) (h = 0.0005) and RK4 (h = 0.0004) for t ∈ [0,10].

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Fig. 2

Dynamics of the Lyapunov exponents with a = 35, b = 3, and c = 28

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Fig. 3

A chaotic case with parameters a = 35, b = 3, and c = 28: cGP(2) (h = 0.0005) versus RK4 (h = 0.0005) and RK4 (h = 0.0001)

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Fig. 4

Phase portraits using cGP(2)-method on h = 0.0005 for a = 35, b = 3, and c = 28

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Fig. 5

A chaotic case with parameters a = 35, b = 3, and c = 28. (Left) Error solutions using cGP(2) (h = 0.0005) and RK4 (h = 0.0005) and (right) error solutions using cGP(2) (h = 0.0005) and RK4 (h = 0.0001) for t ∈ [0,10].

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