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.
Issue Section:
Research Papers
References
1.
Lorenz
, E. N.
, 1963
, “Deterministic Nonperiodic Flow
,” J. Atmos. Sci.
, 20
(2), pp. 130
–141
.10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;22.
Chen
, G.
, and Ueta
, T.
, 1999
, “Yet Another Chaotic Attractor
,” Int. J. Bifurcation Chaos
, 9
(7
), pp. 1465
–1466
.10.1142/S02181274990010243.
Lü
, J.
, Chen
, G.
, and Zhang
, S.
, 2002
, “Dynamical Analysis of a New Chaotic Attractor
,” Int. J. Bifurcation Chaos
, 12
(5
), pp. 1001
–1015
.10.1142/S02181274020048514.
Noorani
, M. S. M.
, Hashim
, I.
, Ahmad
, R.
, Bakar
, S. A.
, Ismail
, E. S.
, and Zakaria
, A. M.
, 2007
, “Comparing Numerical Methods for the Solutions of the Chen System
,” Chaos, Solitons Fractals
32
(4
), pp. 1296
–1304
.10.1016/j.chaos.2005.12.0365.
Hashim
, I.
, Noorani
, M. S. M.
, Ahmad
, R.
, Bakar
, S. A.
, Ismail
, E. S.
, and Zakaria
, A. M.
, 2006
, “Accuracy of the Adomian Decomposition Method Applied to the Lorenz System
,” Chaos, Solitons Fractals
, 28
(5
), pp. 1149
–1158
.10.1016/j.chaos.2005.08.1356.
Liu
, C.
, Liu
, T.
, Liu
, L.
, and Liu
, K.
, 2004
, “A New Chaotic Attractor
,” Chaos, Solitons Fractals
, 22
(5
), pp. 1031
–1038
.10.1016/j.chaos.2004.02.0607.
Gorder
, R. A. V.
, and Choudhury
, S. R.
, 2011
, “Analytical Hopf Bifurcation and Stability Analysis of T System
,” Commun. Theor. Phys.
, 55
(4
), pp. 609–616.10.1088/0253-6102/55/4/178.
Schieweck
, F.
, 2010
, “A-Stable Discontinuous Galerkin–Petrov Time Discretization of Higher Order
,” J. Numer. Math.
, 18
(1
), pp. 25
–57
.10.1515/jnum.2010.0029.
Hussain
, S.
, Schieweck
, F.
, and Turek
, S.
, 2011
, “Higher Order Galerkin Time Discretizations and Fast Multigrid Solvers for the Heat Equation
,” J. Numer. Math.
, 19
(1
), pp. 41
–61
.10.1515/jnum.2011.00310.
Aziz
, A. K.
, and Monk
, P.
, 1989
, “Continuous Finite Elements in Space and Time for the Heat Equation
,” Math. Comput.
, 52
(186
), pp. 255
–274
.10.1090/S0025-5718-1989-0983310-211.
Thomée
, V.
, 2006
, Galerkin Finite Element Methods for Parabolic Problems
(Computational Mathematics), 2nd ed., Vol. 25
, Springer
, Berlin
.10.1007/978-3-662-03359-312.
Kutta
, W.
, 1901
, “Beitrag zur näherungsweisen Integration totaler Differentialgleichungen
,” Z. Math. Phys.
, 46
(1
), pp. 435
–453
.13.
Butcher
, J. C.
, 2008
, Numerical Methods for Ordinary Differential Equations
, Wiley
, Hoboken, NJ
.10.1002/978047075376714.
Matthies
, G.
, and Schieweck
, F.
, 2011
, “Higher Order Variational Time Discretizations for Nonlinear Systems of Ordinary Differential Equations
,” Fakultät für Mathematik
, Otto-von-Guericke Universität Magdeburg
, Magdeburg, Germany.15.
Ueta
, T.
, and Chen
, G.
, 2000
, “Bifurcation Analysis of Chen’s Equation
,” Int. J. Bifurcation Chaos
, 10
(8
), pp. 1917
–1931
.10.1142/S021812740000118316.
Al-sawalha
, M. M.
, and Noorani
, M. S. M.
, 2009
, “A Numeric-Analytic Method for Approximating the Chaotic Chen System
,” Chaos, Solitons Fractals
, 42
(3
), pp. 1784
–1791
.10.1016/j.chaos.2009.03.09617.
Deng
, W.
, and Li
, C.
, 2005
, “Synchronization of Chaotic Fractional Chen System
,” J. Phys. Soc. Jpn.
, 74
(6
), pp. 1645
–1648
.10.1143/JPSJ.74.164518.
Lü
, J.
, Zhou
, T.
, Chen
, G.
, and Zhang
, S.
, 2002
, “Local Bifurcations of the Chen System
,” Int. J. Bifurcation Chaos
, 12
(10
), pp. 2257
–2270
.10.1142/S0218127402005819Copyright © 2015 by ASME
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