Numerical simulation of chaotic dynamical systems by the method of differential quadrature

Lorenz, 1963, Deterministic nonperiodic flow, J. Atmospheric Sci., 20, 130, 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 Sparrow, 1982 Belekolos, 2009, Chaos in a generalized Lorenz system, Chaos Solitons Fractals, 41, 2595, 10.1016/j.chaos.2008.09.049 Chen, 1999, Yet another chaotic attractor, Internat. J. Bifur. Chaos, 9, 1465, 10.1142/S0218127499001024 Ueta, 2000, Bifurcation analysis of Chen’s equation, Internat. J. Bifur. Chaos, 8, 1917, 10.1142/S0218127400001183 Lu, 2002, Local bifurcation of the Chen system, Internat. J. Bifur. Chaos, 12, 2257, 10.1142/S0218127402005819 Yassen, 2002, The optimal control of Chen chaotic dynamical system, Appl. Math. Comput., 171, 171, 10.1016/S0096-3003(01)00137-0 Yassen, 2003, Chaos control of Chen chaotic dynamical system, Chaos Solitons Fractals, 15, 271, 10.1016/S0960-0779(01)00251-X Deng, 2005, Synchronization of chaotic fractional Chen system, J. Phys. Soc. Japan, 74, 1645, 10.1143/JPSJ.74.1645 Plienpanich, 2005, Controllability and stability of the perturbed Chen chaotic dynamical system, Appl. Math. Comput., 171, 927, 10.1016/j.amc.2005.01.099 Chang, 2006, Complex dynamics in Chen’s system, Chaos Solitons Fractals, 27, 75, 10.1016/j.chaos.2004.12.011 C˘elikovsky, 2002, On a generalized Lorenz canonical form of chaotic systems, Internat. J. Bifur. Chaos, 12, 1789, 10.1142/S0218127402005467 Zhou, 2004, Chen’s attractor exists, Internat. J. Bifur. Chaos, 14, 3167, 10.1142/S0218127404011296 Zhou, 2003, Complex dynamical behaviors of the chaotic Chen’s system, Internat. J. Bifur. Chaos, 13, 2561, 10.1142/S0218127403008089 Noorani, 2007, Comparing numerical methods for the solutions of the Chen system, Chaos Solitons Fractals, 32, 1296, 10.1016/j.chaos.2005.12.036 Abdulaziz, 2008, Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos Solitons Fractals, 36, 1405, 10.1016/j.chaos.2006.09.007 Genesio, 1992, A harmonic balance method for the analysis of chaotic dynamics in nonlinear systems, Automatica, 28, 531, 10.1016/0005-1098(92)90177-H Rössler, 1979, An equation for hyperchaotic, Phys. Lett. A, 71, 155, 10.1016/0375-9601(79)90150-6 Zhang, 2005, Controlling and tracking hyperchaotic Rössler system via active back stepping design, Chaos Solitons Fractals, 26, 353, 10.1016/j.chaos.2004.12.032 Mossa Al-sawalha, 2009, On accuracy of Adomian decomposition method for hyperchaotic Rössler system, Chaos Solitons Fractals, 40, 1801, 10.1016/j.chaos.2007.09.062 Guellal, 1997, Numerical study of Lorenz’s equation by the Adomian method, Comut. Math. Appl., 33, 25, 10.1016/S0898-1221(96)00234-9 Vadasz, 2000, Subcritical transitions to chaos and hysteresis in a fluid layer heated from below, Int. J. Heat Mass Transfer, 43, 705, 10.1016/S0017-9310(99)00173-8 Vadasz, 2000, Convergence and accuracy of Adomian’s decomposition method for the solution of Lorenz equations, Int. J. Heat Mass Transfer, 43, 1715, 10.1016/S0017-9310(99)00260-4 Hashim, 2006, Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos Solitons Fractals, 28, 1149, 10.1016/j.chaos.2005.08.135 Allan, 2009, Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method, Chaos Solitons Fractals, 39, 1744, 10.1016/j.chaos.2007.06.116 Goh, 2009, Efficiency of variational iteration method for chaotic Genesio system—Classical and multistage approach, Chaos Solitons Fractals, 40, 2152, 10.1016/j.chaos.2007.10.003 Chowdhury, 2009, The multistage homotopy-perturbation method: a powerful scheme for handling the Lorenz system, Chaos Solitons Fractals, 40, 1929, 10.1016/j.chaos.2007.09.073 Chowdhury, 2009, Application of multistage homotopy-perturbation method for the solutions of the Chen system, Nonlinear Anal. RWA, 10, 381, 10.1016/j.nonrwa.2007.09.014 Mossa Al-sawalha, 2009, A numeric-analytic method for approximating the chaotic Chen system, Chaos Solitans Fractals, 42, 1784, 10.1016/j.chaos.2009.03.096 He, 2000, A coupling method of homotopy technique and perturbation technique for nonlinear problems, Internat. J. Non-Linear Mech., 35, 37, 10.1016/S0020-7462(98)00085-7 He, 2005, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 26, 695, 10.1016/j.chaos.2005.03.006 Bellman, 1971, Differential quadrature and long term integrations, J. Math. Anal. Appl., 34, 235, 10.1016/0022-247X(71)90110-7 Bellman, 1972, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Comput. Phys., 10, 40, 10.1016/0021-9991(72)90089-7 Shu, 2000 Wu, 2000, The generalized differential quadrature rule for initial-value differential equations, J. Sound Vib., 233, 195, 10.1006/jsvi.1999.2815 Wu, 2000, Numerical solution for differential equations of Duffing-type non-linearity using the generalized differential quadrature, J. Sound Vib., 237, 805, 10.1006/jsvi.2000.3050 Shu, 2002, Block-marching in time with DQ discretization: an efficient method for time-dependent problems, Comput. Methods Appl. Mech. Eng., 191, 4587, 10.1016/S0045-7825(02)00387-0 Tanaka, 2001, Coupling dual reciprocity BEM and differential quadrature method for time-dependent diffusion problems, Appl. Math. Model., 25, 257, 10.1016/S0307-904X(00)00052-4 Tanaka, 2001, Dual reciprocity BEM applied to transient elastodynamic problems with differential quadrature method in time, Comput. Methods Appl. Mech. Eng., 190, 2331, 10.1016/S0045-7825(00)00237-1 Fung, 2001, Solving initial value problems by differential quadrature method—Part 1: first-order equations, Internat. J. Numer. Methods Engrg., 50, 1411, 10.1002/1097-0207(20010228)50:6<1411::AID-NME78>3.0.CO;2-O Fung, 2001, Solving initial value problems by differential quadrature method—Part 2: second- and higher-order equations, Internat. J. Numer. Methods Engrg., 50, 1429, 10.1002/1097-0207(20010228)50:6<1429::AID-NME79>3.0.CO;2-A Fung, 2002, Stability and accuracy of differential quadrature method in solving dynamic problems, Comput. Methods Appl. Mech. Eng., 191, 1311, 10.1016/S0045-7825(01)00324-3 Zhong, 2006, Solution of nonlinear initial-value problems by the spline-based differential quadrature method, J. Sound Vib., 296, 908, 10.1016/j.jsv.2006.03.018 Liu, 2008, An assessment of the differential quadrature time integration scheme for nonlinear dynamic equations, J. Sound Vib., 314, 246, 10.1016/j.jsv.2008.01.004 Eftekhari, 2009, Dynamic analysis of laminated composite coated beams carrying multiple accelerating oscillators using a coupled finite element-differential quadrature method, ASME J. Appl. Mech., 76, 061001, 10.1115/1.3114969 Eftekhari, 2010, A coupled finite element-differential quadrature element method and its accuracy for moving load problem, Appl. Math. Model., 34, 228, 10.1016/j.apm.2009.03.039 Khalili, 2010, A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads, Compos. Struct., 92, 2497, 10.1016/j.compstruct.2010.02.012 Jafari, 2011, A new mixed finite element-differential quadrature formulation for forced vibration of beams carrying moving loads, ASME J. Appl. Mech., 78, 011020, 10.1115/1.4002037 Eftekhari, 2012, Coupling Ritz method and triangular quadrature rule for moving mass problem, ASME J. Appl. Mech., 79, 021018, 10.1115/1.4005577 Quan, 1989, New insights in solving distributed system equations by the quadrature methods, Part I: analysis, Comput. Chem. Eng., 13, 779, 10.1016/0098-1354(89)85051-3 Bert, 1996, Differential quadrature method in computational mechanics: a review, ASME J. Appl. Mech. Rev., 49, 1, 10.1115/1.3101882