Chaos analysis of the nonlinear duffing oscillators based on the new Adomian polynomials

Volume 9, Issue 4, pp 1877--1881 http://dx.doi.org/10.22436/jnsa.009.04.41

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Authors

L. L. Huang - Institute of Applied Nonlinear Science, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China. G. C. Wu - Institute of Applied Nonlinear Science, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China. M. M. Rashidi - Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Address: 4800 Cao An Rd., Jiading, Shanghai 201804, China. - ENN-Tongji Clean Energy Institute of Advanced Studies, China. W. H. Luo - Institute of Applied Nonlinear Science, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China.

Abstract

Numerical recurrence formulae are given to investigate the chaotic motion of the famous Duffing system. The new Adomian polynomial is adopted to treat the cubic nonlinear term. With the numerical simulation of the phase portraits and the Poincare sections, the chaotic behaviors are discussed for varied frequencies, damping coefficients and forces. The results show that the numerical method is reliable to investigate chaotic systems.

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Keywords

• New Adomian polynomial
• duffing systems
• chaos.

MSC

•  47H10
•  54H25

References

• [1] G. Adomian, Solving frontier problems of physics: the decomposition method, Kluwer Academic Publishers Group, Dordrecht (1994)


• [2] V. Daftardar-Gejji, H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301 (2005), 508-518.


• [3] J. S. Duan , An efficient algorithm for the multivariable Adomian polynomials, Appl. Math. Comput., 217 (2010), 2456-2467.


• [4] J. S. Duan, Recurrence triangle for Adomian polynomials, Appl. Math. Comput., 216 (2010), 1235-1241.


• [5] J. S. Duan, Convenient analytic recurrence algorithms for the Adomian polynomials, Appl. Math. Comput., 217 (2011), 6337-6348.


• [6] J. S. Duan, R. Rach, New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods, Appl. Math. Comput., 218 (2011), 2810-2828.


• [7] J. S. Duan, R. Rach, A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl. Math. Comput., 218 (2011), 4090-4118.


• [8] G. Erjaee, Application of the multistage homotopy perturbation method to some dynamical systems, Iran. J. Sci. Technol. Trans. A Sci., 35 (2011), 33-38.


• [9] L. L. Huang , Transformation of series-A case of the exponent functions, J. Comput. Complex. Appl., 2 (2016), 44-45.


• [10] S. W. Huang, T. Z. Zhen, Visualization based Analysis of Dynamic Property of Chaos in nonlinear forced vibration, J. Hohai Univ. (Natural Science Edition), 30 (2002), 42-46.


• [11] I. Kovacic, M. J. Brennan, The Duffing equation: nonlinear oscillators and their behaviour, John Wiley & Sons, Chichester (2011)


• [12] C. Li, G. Peng, Chaos in Chen's system with a fractional order, Chaos Solitons Fractals, 22 (2004), 443-450.


• [13] S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488-494.


• [14] U. Parlitz, W. Lauterborn, Superstructure in the bifurcation set of the Duffing equation \(\frac{dx^2}{ dt^2} +\frac{ dx}{ dt} + x + x^3 = fcos(wt)\) , Phys. Lett. A, 107 (1985), 351-355.


• [15] P. Y. Tsai, C. K. Chen, An approximate analytic solution of the nonlinear Riccati differential equation, J. Franklin Inst., 347 (2010), 1850-1862.


• [16] A. M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102 (1999), 77-86.


• [17] G. C. Wu, D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dynam., 75 (2014), 283-287.


• [18] X. Wu, J. Li, G. Chen, Chaos in the fractional order unified system and its synchronization , J. Franklin Inst., 345 (2008), 392-401.


• [19] Y. Zeng, Approximate solutions of three integral equations by the new Adomian decomposition method, J. Comput. Complex. Appl., 2 (2016), 38-43.