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Year 2022, Volume: 51 Issue: 1, 27 - 47, 14.02.2022
https://doi.org/10.15672/hujms.885150

Abstract

References

  • [1] O. Abdulaziz, N. Noor, I. Hashim, and M. Noorani, Further accuracy tests on adomian decomposition method for chaotic systems, Chaos, Solitons & Fractals 36, 1405–1411, 2008.
  • [2] M.E. Adiyaman and S. Somali, Taylor’s decomposition on two points for onedimensional Bratu problem, Numer. Methods Partial Differential Equations, 26, 412– 425, 2010.
  • [3] M.E. Adiyaman and S. Somali, A new approach for linear eigenvalue problems and nonlinear Euler buckling problem, Abstr. Appl. Anal. 2012, Article ID 697013, 2012.
  • [4] A.K. Alomari, M.S.M. Noorani, and R. Nazar, Homotopy approach for the hyperchaotic Chen system, Phys. Scr. 8, 045005, 2010.
  • [5] A.K. Alomari, M.S.M. Noorani and R. Nazar, Adaptation of homotopy analysis method for the numeric analytic solution of Chen system, Commun. Nonlinear Sci. Numer. Simul. 14, 2336-2346, 2009.
  • [6] A. Ashyralyev and P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Operator Theory Advances and Applications, Birkhauser, Basel, 2004.
  • [7] B. Batiha, M.S.M. Noorani, I. Hashim, and E.S. Ismail, The multistage variational iteration method for a class of nonlinear system of odes, Phys. Scr. 76, 388–392, 2007.
  • [8] D.N. Butusov, V.S. Andreev, and D.O. Pesterev, Composition semi-implicit methods for chaotic problems simulation, in: 2016 XIX IEEE International Conference on Soft Computing and Measurements (SCM), 107–110, 2016.
  • [9] D.N. Butusov, V.Y. Ostrovskii, A.I. Karimov, and V.S. Andreev, Semi-explicit composition methods in memcapacitor circuit simulation, Int. J. Embed. Real-Time Commun. Syst. 10(2), 37–52, 2019.
  • [10] D.N. Butusov, A. Tutueva, P. Fedoseev, A. Terentev, and A. Karimov, Semi-implicit multistep extrapolation ODE solvers, Mathematics 8(6), 943, 2020.
  • [11] M.S.H. Chowdhury, I. Hashim and S. Momani, The multistage homotopy-perturbation method: a powerful scheme for handling the Lorenz system, Chaos, Solitons & Fractals 40, 1929–1937, 2009.
  • [12] M.S.H. Chowdhury, I. Hashim, S. Momani, and M.M. Rahman, Application of multistage homotopy perturbation method to the chaotic Genesio system, Abstr. Appl. Anal. 2012, Article ID 974293, 2012.
  • [13] M.S.H. Chowdhury, I. Hashim, Application of multistage homotopy-perturbation method for the solutions of the Chen system, Nonlinear Anal.: Real World Appl. 10, 381–391, 2009.
  • [14] Y. D, B. Jang, Enhanced multistage differential transform method: application to the population models, Abstr. Appl. Anal. 2012, Article ID 253890, 2012.
  • [15] A. Freihat, S. Momani, Adaptation of differential transform method for the numeric-analytic solution of fractional-order Rössler chaotic and hyperchaotic systems, Abstr. Appl. Anal. 2012, Article ID 934219, 2012.
  • [16] A. Ghorbani, J. Saberi-Nadja, A piecewise-spectral parametric iteration method for solving the non-linear chaotic Genesio system, Math. Comput. Model. 54, 131–139, 2011.
  • [17] S.M. Goh, M.S.M. Noorani, and I. Hashim, Efficacy of variational iteration method for chaotic Genesio systemclassical and multistage approach, Chaos, Solitons & Fractals 40, 2152–2159, 2009.
  • [18] S.M. Goh, M.S.M. Noorani, I. Hashim, and M.M. Al-Sawalha, Variational iteration method as a reliable treatment for the hyperchaotic Rössler system, Int. J. Nonlinear Sci. Numer. Simul. 10, 363–371, 2009.
  • [19] G. González-Parra, A.J. Arenas, and L. Jódar, Piecewise infinite series solutions of seasonal diseases models using multistage Adomian method, Commun. Nonlinear Sci. Numer. Simul. 14, 3967–3977, 2009.
  • [20] E. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130–141, 1963.
  • [21] R. Lozi, V.A. Pogonin, and A.N. Pchelintsev, A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities, Chaos, Solitons & Fractals 91, 108–114, 2016.
  • [22] D. Mathale, P.G. Dlamini, and M. Khumalo, Compact finite difference relaxation method for chaotic and hyperchaotic initial value systems, Comp. Appl. Math. 37, 5187–5202, 2018.
  • [23] M. Mossa Al-Sawalha, M.S.M. Noorani, and I.Hashim, On accuracy of Adomian decomposition method for hyperchaotic Rössler system, Chaos, Solitons & Fractals 40, 1801–1807, 2009.
  • [24] S.S. Motsa, P. Dlamini, and M. Khumalo, A new multistage spectral relaxation method for solving chaotic initial value systems, Nonlinear Dyn. 72, 265–283, 2013.
  • [25] S.S. Motsa, Y. Khan, and S. Shateyi, Application of piecewise successive linearization method for the solutions of the chen chaotic system, J. Appl. Math 2012, Article ID 258948, 2012.
  • [26] S.S. Motsa, A new piecewise-quasilinearization method for solving chaotic systems of initial value problems, Cent. Eur. J. Phys. 10, 936–946, 2012.
  • [27] Z.M. Odibat, C. Bertelle, M.A. Aziz-Alaoui, and G.H.E. Duchamp, A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. Math. Appl. 59, 1462–1472, 2010.
  • [28] L.M. Resler, Edward N. Lorenz’s 1963 paper, "Deterministic nonperiodic flow", in Journal of the Atmospheric Sciences, Vol 20, pages 130-141: Its history and relevance to physical geography, Prog Phys Geogr . Earth Environ. 40, 175–180, 2015.
  • [29] O.E. Rössler, An equation for continuous chaos, Phys. Lett. A 57, 397–398, 1976.
  • [30] M. Turkyilmazoglu, A simple algorithm for high order Newton iteration formulae and some new variants, Hacet. J. Math. Stat. 49(1), 425–438, 2020.
  • [31] M. Turkyilmazoglu, Is homotopy perturbation method the traditional Taylor series expansion, Hacet. J. Math. Stat. 44(3), 651–657, 2015.
  • [32] M. Turkyilmazoglu, Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer, Phys. Fluids 21, 106104, 2009.
  • [33] A. Tutueva, T. Karimov, and D.N. Butusov, Semi-implicit and semi-explicit Adams- Bashforth-Moulton methods, Mathematics 8(5), 780, 2020.
  • [34] S. Wang, Y. Yu, Application of multistage homotopyperturbation method for the solutions of the chaotic fractional order systems, Int. J. Nonlinear Sci. 13, 3–14, 2012.
  • [35] L. Yao, Computed chaos or numerical errors, Nonlinear Anal. Model. Control 15, 109–126, 2010.

High order approach for solving chaotic and hyperchaotic problems

Year 2022, Volume: 51 Issue: 1, 27 - 47, 14.02.2022
https://doi.org/10.15672/hujms.885150

Abstract

In this work, the method of Taylor's decomposition on two points is suggested in order to find approximate solutions of chaotic and hyperchaotic initial value problems and to analyze the behaviors of these solutions. Unlike to the classical Taylor's method, the proposed numerical scheme is based on the application of the Taylor's decomposition on two points to the system of nonlinear initial value problems, and as a result an implicit method is obtained. Stability and error analysis of the method are presented, and its high-order accuracy and A-stability are proven. One of the advantages of the proposed method is that it is a stable and very efficient method for chaotic problems as it is an implicit one-step method. The most important advantage of the Taylor's decomposition method is that it has high order accuracy for large step sizes with a simple algorithm compared to other methods. The applicability of the proposed method has been examined in some famous chaotic systems; the Lorenz and Chen systems, and hyperchaotic systems; the Chua and Rabinovich-Fabrikant systems, to emphasize both its accuracy and effectiveness. The accuracy of the proposed method is checked by comparing the calculated results with semi-explicit Adams-Bashforth-Moulton method and ninth order Runge-Kutta method. The calculated results are also compared with multi-stage spectral relaxation method and multi-domain compact finite difference relaxation method. Comparisons have shown that the method is more accurate and efficient than the other mentioned methods for large step sizes. The obtained results are also compared with the theoretical findings and it is shown that the theoretical and numerical results are consistent.

References

  • [1] O. Abdulaziz, N. Noor, I. Hashim, and M. Noorani, Further accuracy tests on adomian decomposition method for chaotic systems, Chaos, Solitons & Fractals 36, 1405–1411, 2008.
  • [2] M.E. Adiyaman and S. Somali, Taylor’s decomposition on two points for onedimensional Bratu problem, Numer. Methods Partial Differential Equations, 26, 412– 425, 2010.
  • [3] M.E. Adiyaman and S. Somali, A new approach for linear eigenvalue problems and nonlinear Euler buckling problem, Abstr. Appl. Anal. 2012, Article ID 697013, 2012.
  • [4] A.K. Alomari, M.S.M. Noorani, and R. Nazar, Homotopy approach for the hyperchaotic Chen system, Phys. Scr. 8, 045005, 2010.
  • [5] A.K. Alomari, M.S.M. Noorani and R. Nazar, Adaptation of homotopy analysis method for the numeric analytic solution of Chen system, Commun. Nonlinear Sci. Numer. Simul. 14, 2336-2346, 2009.
  • [6] A. Ashyralyev and P.E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Operator Theory Advances and Applications, Birkhauser, Basel, 2004.
  • [7] B. Batiha, M.S.M. Noorani, I. Hashim, and E.S. Ismail, The multistage variational iteration method for a class of nonlinear system of odes, Phys. Scr. 76, 388–392, 2007.
  • [8] D.N. Butusov, V.S. Andreev, and D.O. Pesterev, Composition semi-implicit methods for chaotic problems simulation, in: 2016 XIX IEEE International Conference on Soft Computing and Measurements (SCM), 107–110, 2016.
  • [9] D.N. Butusov, V.Y. Ostrovskii, A.I. Karimov, and V.S. Andreev, Semi-explicit composition methods in memcapacitor circuit simulation, Int. J. Embed. Real-Time Commun. Syst. 10(2), 37–52, 2019.
  • [10] D.N. Butusov, A. Tutueva, P. Fedoseev, A. Terentev, and A. Karimov, Semi-implicit multistep extrapolation ODE solvers, Mathematics 8(6), 943, 2020.
  • [11] M.S.H. Chowdhury, I. Hashim and S. Momani, The multistage homotopy-perturbation method: a powerful scheme for handling the Lorenz system, Chaos, Solitons & Fractals 40, 1929–1937, 2009.
  • [12] M.S.H. Chowdhury, I. Hashim, S. Momani, and M.M. Rahman, Application of multistage homotopy perturbation method to the chaotic Genesio system, Abstr. Appl. Anal. 2012, Article ID 974293, 2012.
  • [13] M.S.H. Chowdhury, I. Hashim, Application of multistage homotopy-perturbation method for the solutions of the Chen system, Nonlinear Anal.: Real World Appl. 10, 381–391, 2009.
  • [14] Y. D, B. Jang, Enhanced multistage differential transform method: application to the population models, Abstr. Appl. Anal. 2012, Article ID 253890, 2012.
  • [15] A. Freihat, S. Momani, Adaptation of differential transform method for the numeric-analytic solution of fractional-order Rössler chaotic and hyperchaotic systems, Abstr. Appl. Anal. 2012, Article ID 934219, 2012.
  • [16] A. Ghorbani, J. Saberi-Nadja, A piecewise-spectral parametric iteration method for solving the non-linear chaotic Genesio system, Math. Comput. Model. 54, 131–139, 2011.
  • [17] S.M. Goh, M.S.M. Noorani, and I. Hashim, Efficacy of variational iteration method for chaotic Genesio systemclassical and multistage approach, Chaos, Solitons & Fractals 40, 2152–2159, 2009.
  • [18] S.M. Goh, M.S.M. Noorani, I. Hashim, and M.M. Al-Sawalha, Variational iteration method as a reliable treatment for the hyperchaotic Rössler system, Int. J. Nonlinear Sci. Numer. Simul. 10, 363–371, 2009.
  • [19] G. González-Parra, A.J. Arenas, and L. Jódar, Piecewise infinite series solutions of seasonal diseases models using multistage Adomian method, Commun. Nonlinear Sci. Numer. Simul. 14, 3967–3977, 2009.
  • [20] E. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130–141, 1963.
  • [21] R. Lozi, V.A. Pogonin, and A.N. Pchelintsev, A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities, Chaos, Solitons & Fractals 91, 108–114, 2016.
  • [22] D. Mathale, P.G. Dlamini, and M. Khumalo, Compact finite difference relaxation method for chaotic and hyperchaotic initial value systems, Comp. Appl. Math. 37, 5187–5202, 2018.
  • [23] M. Mossa Al-Sawalha, M.S.M. Noorani, and I.Hashim, On accuracy of Adomian decomposition method for hyperchaotic Rössler system, Chaos, Solitons & Fractals 40, 1801–1807, 2009.
  • [24] S.S. Motsa, P. Dlamini, and M. Khumalo, A new multistage spectral relaxation method for solving chaotic initial value systems, Nonlinear Dyn. 72, 265–283, 2013.
  • [25] S.S. Motsa, Y. Khan, and S. Shateyi, Application of piecewise successive linearization method for the solutions of the chen chaotic system, J. Appl. Math 2012, Article ID 258948, 2012.
  • [26] S.S. Motsa, A new piecewise-quasilinearization method for solving chaotic systems of initial value problems, Cent. Eur. J. Phys. 10, 936–946, 2012.
  • [27] Z.M. Odibat, C. Bertelle, M.A. Aziz-Alaoui, and G.H.E. Duchamp, A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. Math. Appl. 59, 1462–1472, 2010.
  • [28] L.M. Resler, Edward N. Lorenz’s 1963 paper, "Deterministic nonperiodic flow", in Journal of the Atmospheric Sciences, Vol 20, pages 130-141: Its history and relevance to physical geography, Prog Phys Geogr . Earth Environ. 40, 175–180, 2015.
  • [29] O.E. Rössler, An equation for continuous chaos, Phys. Lett. A 57, 397–398, 1976.
  • [30] M. Turkyilmazoglu, A simple algorithm for high order Newton iteration formulae and some new variants, Hacet. J. Math. Stat. 49(1), 425–438, 2020.
  • [31] M. Turkyilmazoglu, Is homotopy perturbation method the traditional Taylor series expansion, Hacet. J. Math. Stat. 44(3), 651–657, 2015.
  • [32] M. Turkyilmazoglu, Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer, Phys. Fluids 21, 106104, 2009.
  • [33] A. Tutueva, T. Karimov, and D.N. Butusov, Semi-implicit and semi-explicit Adams- Bashforth-Moulton methods, Mathematics 8(5), 780, 2020.
  • [34] S. Wang, Y. Yu, Application of multistage homotopyperturbation method for the solutions of the chaotic fractional order systems, Int. J. Nonlinear Sci. 13, 3–14, 2012.
  • [35] L. Yao, Computed chaos or numerical errors, Nonlinear Anal. Model. Control 15, 109–126, 2010.
There are 35 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Meltem Adıyaman 0000-0001-5065-7068

Publication Date February 14, 2022
Published in Issue Year 2022 Volume: 51 Issue: 1

Cite

APA Adıyaman, M. (2022). High order approach for solving chaotic and hyperchaotic problems. Hacettepe Journal of Mathematics and Statistics, 51(1), 27-47. https://doi.org/10.15672/hujms.885150
AMA Adıyaman M. High order approach for solving chaotic and hyperchaotic problems. Hacettepe Journal of Mathematics and Statistics. February 2022;51(1):27-47. doi:10.15672/hujms.885150
Chicago Adıyaman, Meltem. “High Order Approach for Solving Chaotic and Hyperchaotic Problems”. Hacettepe Journal of Mathematics and Statistics 51, no. 1 (February 2022): 27-47. https://doi.org/10.15672/hujms.885150.
EndNote Adıyaman M (February 1, 2022) High order approach for solving chaotic and hyperchaotic problems. Hacettepe Journal of Mathematics and Statistics 51 1 27–47.
IEEE M. Adıyaman, “High order approach for solving chaotic and hyperchaotic problems”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, pp. 27–47, 2022, doi: 10.15672/hujms.885150.
ISNAD Adıyaman, Meltem. “High Order Approach for Solving Chaotic and Hyperchaotic Problems”. Hacettepe Journal of Mathematics and Statistics 51/1 (February 2022), 27-47. https://doi.org/10.15672/hujms.885150.
JAMA Adıyaman M. High order approach for solving chaotic and hyperchaotic problems. Hacettepe Journal of Mathematics and Statistics. 2022;51:27–47.
MLA Adıyaman, Meltem. “High Order Approach for Solving Chaotic and Hyperchaotic Problems”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, 2022, pp. 27-47, doi:10.15672/hujms.885150.
Vancouver Adıyaman M. High order approach for solving chaotic and hyperchaotic problems. Hacettepe Journal of Mathematics and Statistics. 2022;51(1):27-4.