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A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems

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Abstract

Nonlinear chaotic systems yield many interesting features related to different physical phenomena and practical applications. These systems are very sensitive to initial conditions at each time-iteration level in a numerical algorithm. In this article, we study the behavior of some nonlinear chaotic systems by a new numerical approach based on the concept of Galerkin–Petrov time-discretization formulation. Computational algorithms are derived to calculate dynamical behavior of nonlinear chaotic systems. Dynamical systems representing weather prediction model and finance model are chosen as test cases for simulation using the derived algorithms. The obtained results are compared with classical RK-4 and RK-5 methods, and an excellent agreement is achieved. The accuracy and convergence of the method are shown by comparing numerically computed results with the exact solution for two test problems derived from another nonlinear dynamical system in two-dimensional space. It is shown that the derived numerical algorithms have a great potential in dealing with the solution of nonlinear chaotic systems and thus can be utilized to delineate different features and characteristics of their solutions.

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References

  1. Luo, S., Sun, Q., Cheng, W.: Chaos control of the micro-electro-mechanical resonator by using adaptive dynamic surface technology with extended state observer. AIP Adv. 6, 045104 (2016)

    Article  Google Scholar 

  2. Liu, Y., Zhang, J.: Predicting traffic flow in local area networks by the largest Lyapunov exponent. Entropy 18(1), 32 (2016)

    Article  Google Scholar 

  3. Lim, Y., Han, J., Lam, Y.C.: Chaos analysis of viscoelastic chaotic flows of polymeric fluids in a micro-channel. AIP Adv. 5, 077150 (2015)

    Article  Google Scholar 

  4. Baskonus, H.M., Mekkaoui, T., Hammouch, Z., Bulut, H.: Active control of a chaotic fractional order economic system. Entropy 17(8), 5771–5783 (2015)

    Article  Google Scholar 

  5. Wang, Y., Wong, K., Liao, X., Chen, G.: A new chaos-based fast image encryption algorithm. Appl. Soft Comput. 11(1), 514–522 (2011)

    Article  Google Scholar 

  6. Wang, X., Teng, L., Qin, X.: A novel color image encryption algorithm based on chaos. Signal Process 92, 1101–1108 (2012)

    Article  Google Scholar 

  7. Liu, H.J., Wang, X.Y.: Color image encryption based on one-time keys and robust chaotic maps. Comput. Math. Appl. 59, 3320–3327 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wang, X.Y., Zhang, Y.Q., Bao, X.M.: A color image encryption scheme using permutation-substitution based on chaos. Entropy 17(6), 3877–3897 (2015)

    Article  Google Scholar 

  9. Kikuchi, R., Yano, T., Tsutsumi, A., Yoshida, K., Punchochar, M., Drahos, J.: Diagnosis of chaotic dynamics of bubble motion in a bubble column. Chem. Eng. Sci. 52(21–22), 3741 (1997)

    Article  Google Scholar 

  10. Letzel, H.M., Schouten, J.C., Krishna, R., van den Bleek, C.M.: Characterization of regimes and regime transitions in bubble columns by chaos analysis of pressure signals. Chem. Eng. Sci. 52(24), 4447 (1997)

    Article  Google Scholar 

  11. Guzmán, A.M., Amon, C.H.: Dynamical flow characterization of transitional and chaotic regimes in converging–diverging channels. J. Fluid Mech. 321, 25–57 (1996)

    Article  MATH  Google Scholar 

  12. Pine, D.J., Gollub, J.P., Brady, J.F., Leshansky, A.M.: Chaos and threshold for irreversibility in sheared suspension. Nature 438(7070), 997 (2005)

    Article  Google Scholar 

  13. Leonov, G.A., Kuznetsov, N.V.: On differences and similarities in the analysis of Lorenz, Chen and Lu systems. Appl. Math. Comput. 256, 334–343 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Gao, Q., Ma, J.: Chaos and Hopf bifurcation of a finance system. Nonlinear Dyn. 58, 209–216 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, D.: Global dynamics in the periodically-forced-Chen system. Prog. Theor. Phys. 112(5), 785–796 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eftekhari, S.A., Nejad, F.B., Dowell, E.H.: Bifurcation boundary analysis as a nonlinear damage detection feature: does it work? J. Fluids Struct. 27(2), 297–310 (2011)

    Article  Google Scholar 

  17. Ma, R., Ban, X., Pang, J.S., Liu, H.X.: Submission to the DTA2012 special issue: convergence of time discretization schemes for continuous-time dynamic network loading models. Netw. Spatial Econ. 15(3), 419–441 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Song, C., Qiao, Y.: A novel image encryption algorithm based on DNA encoding and spatiotemporal chaos. Entropy 17(10), 6954–6968 (2015)

    Article  MathSciNet  Google Scholar 

  19. Dehghan, M., Shokri, A.: A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn. 50, 111–120 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, C.C.: Application of a hybrid numerical method to the nonlinear dynamic analysis of a micro gas bearing system. Nonlinear Dyn. 59, 695–710 (2010)

    Article  MATH  Google Scholar 

  21. Shakeri, F., Dehghan, M.: Numerical solution of the Klein–Gordon equation via He’s variational iteration method. Nonlinear Dyn. 51, 89–97 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, X.Y., Lei, Y., Liu, R., Kadir, A.: A chaotic image encryption algorithm based on perceptron model. Nonlinear Dyn. 62, 615–621 (2010)

    Article  MATH  Google Scholar 

  23. Jamal, M., Braikat, B., Boutmir, S., Damil, N., Potier-Ferry, M.: A high order implicit algorithm for solving instationary non-linear problems. Comput. Mech. 28(375–380), 34 (2002)

    MathSciNet  MATH  Google Scholar 

  24. Wang, C.C.: Application of a hybrid numerical method to the nonlinear dynamic analysis of a micro gas bearing system. Nonlinear Dyn. 59, 695–710 (2010)

    Article  MATH  Google Scholar 

  25. Sprott, J.C.: Chaos and Time-Series Analysis. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  26. Drahos, J., Tihon, J., Serio, C., Lubbert, A.: Deterministic chaos analysis of pressure fluctuations in a horizontal pipe at intermittent flow regime. Chem. Eng. J. Biochem. Eng. J. 64(1), 149 (1996)

    Article  Google Scholar 

  27. Wang, S.F., Mosdorf, R., Shoji, M.: Nonlinear analysis on fluctuation feature of two-phase flow through a T-junction. Int. J. Heat Mass Transf. 46(9), 1519 (2003)

    Article  Google Scholar 

  28. Kuzmin, D., ämäläinen, J.H.: Finite Element Methods for Computational Fluid Dynamics. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  29. Fambri, F., Dumbser, M.: Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids. Comput. Methods Appl. Mech. Eng. 324, 170–203 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Guo, R., Filbet, F., Xu, Y.: Efficient high order semi-implicit time discretization and local discontinuous Galerkin methods for highly nonlinear PDEs. J. Sci. Comput. 68(3), 1029–1054 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dorodnitsyn, A.V., Bisnovatyi-Kogan, G.S.: Application of the Galerkin method to the problem of stellar stability. AIP Conf. Proc. 586, 760 (2001)

    Article  MATH  Google Scholar 

  32. Motsa, S.S.: A new piecewise-quasilinearization method for solving chaotic systems of initial value problems. Cent. Eur. J. Phys. 10(4), 936–946 (2012)

    Google Scholar 

  33. Pchelintsev, A.N.: Numerical and physical modeling of the dynamics of the Lorenz system. Numer. Anal. Appl. 7(2), 159–167 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tamma, K.K., Zhou, X., Sha, D.: The time dimension: a theory towards the evolution, classification, characterization and design of computational algorithms for transient/dynamic applications. Arch. Comput. Methods Eng. 7(2), 67–290 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Viswanath, D.: The fractal property of the Lorenz attractor. Physica D 190, 115–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author is grateful to Higher Education Commission (HEC) of Pakistan for providing financial support under the research Grant Numbers 21-557/SRGP/R&D/HEC/2014, in order to complete this research work.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad, 44000, Pakistan

    Muhammad Sabeel Khan

  2. Department of Mechanical Engineering, Khawaja Fareed University of Engineering and Information Technology, Abu Dhabi Road, Rahim Yar Khan, Pakistan

    Muhammad Ijaz Khan

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  1. Muhammad Sabeel Khan
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  2. Muhammad Ijaz Khan
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Correspondence to Muhammad Sabeel Khan.

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Khan, M.S., Khan, M.I. A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems. Nonlinear Dyn 91, 1555–1569 (2018). https://doi.org/10.1007/s11071-017-3964-5

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  • DOI: https://doi.org/10.1007/s11071-017-3964-5

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