Journal of Applied Nonlinear Dynamics
An Accurate Numerical Method and Algorithm for Constructing Solutions of Chaotic Systems
Journal of Applied Nonlinear Dynamics 9(2) (2020) 207221  DOI:10.5890/JAND.2020.06.004
Alexander N. Pchelintsev
Department of Higher Mathematics, Tambov State Technical University, ul. Sovetskaya 106, Tambov, 392000, Russia
Download Full Text PDF
Abstract
In various fields of natural science, the chaotic systems of differential equations are considered more than 50 years. The correct prediction of the behaviour of solutions of dynamical model equations is important in understanding of evolution process and reduce uncertainty. However, often used numerical methods are unable to do it on large time segments. In this article, the author considers the modern numerical method and algorithm for constructing solutions of chaotic systems on the example of tumor growth model. Also a modification of Benettin’s algorithm presents for calculation of Lyapunov exponents.
Acknowledgments
The reported study was funded by RFBR according to the research project 200100347.
References

[1]  Lorenz, E.N. (1963), Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20(2), 130141. 

[2]  Nemytskii, V.V. and Stepanov, V.V. (1989), Qualitative Theory of Differential Equations, Dover Publications: New York. 

[3]  Guckenheimer, J. (1976), A Strange, Strange attractor, in the Hopf bifurcation and its application, Applied Mathematical Series, 19, 368381. 

[4]  Afraimovich, V.S., Bykov, V.V., and Shilnikov, L.P. (1977), The origin and structure of the Lorenz attractor, Soviet Physics Doklady, 22, 253255. 

[5]  Williams, R.F. (1979), The structure of Lorenz attractors, Publications mathématiques de l'IHÉS, 50, 321 347. 

[6]  Kaplan, J.L. and Yorke, J.A. (1979), Preturbulence: a regime observed in a fluid flow model of Lorenz, Communications in Mathematical Physics, 6(2), 93108. 

[7]  Cook, A.E. and Roberts P.H. (1970), The Rikitake twodisc dynamo system, Mathematical Proceedings of the Cambridge Philosophical Society, 68(2), 547569. 

[8]  Tyson, J.J. (1977), On the appearance of chaos in a model of the Belousov reaction, Journal of Mathematical Biology, 5(4), 351362. 

[9]  Vallis, G.K. (1986), El Ni˜no: A chaotic dynamical system? Science, 232(4747), 243245. 

[10]  Vallis, G.K. (1988), Conceptual models of El Ni˜ño and the Southern Oscillation, Journal of Geophysical Research, 93(C11), 1397913991. 

[11]  Sprott, J.C. (1994), Some simple chaotic flows, Physical Review E, 50(2), R647. 

[12]  Sprott, J.C. (1997), Simplest dissipative chaotic flow, Physics Letters A, 228(45), 271274. 

[13]  Wei, Z. (2011), Dynamical behaviors of a chaotic system with no equilibria, Physics Letters A, 376(2), 102108. 

[14]  Wang, X., Chen, G. (2012), A chaotic system with only one stable equilibrium, Communications in Nonlinear Science and Numerical Simulation, 17(3), 12641272. 

[15]  Stenflo, L. (1996), Generalized Lorenz equations for acousticgravity waves in the atmosphere, Physica Scripta, 53(1), 8384. 

[16]  Chen, G. and Ueta, T. (1999), Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9(7), 14651466. 

[17]  Ueta, T. and Chen, G. (2000), Bifurcation analysis of Chen’s equation, International Journal of Bifurcation and Chaos, 10(8), 19171931. 

[18]  Magnitskii, N.A. and Sidorov, S.V. (2006), New Methods for Chaotic Dynamics, World Scientific: Singapore. 

[19]  Afraimovich, V., Gong, X. and Rabinovich, M. (2015), Sequential memory: binding dynamics, Chaos, 25, 103118. 

[20]  Rabinovich, M.I., Afraimovich, V.S. and Varona, P. (2010), Heteroclinic binding, Dynamical Systems, 25(3), 433442. 

[21]  Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A. and Prasad, A. (2016), Hidden attractors in dynamical systems, Physics Reports, 637(3), 150. 

[22]  Yorke, J.A. and Yorke, E.D. (1979), Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model, Journal of Statistical Physics, 21(3), 263277. 

[23]  Yao, L.S. (2010), Computed chaos or numerical errors, Nonlinear Analysis: Modelling and Control, 15(1), 109126. 

[24]  Sparrow, C. (1982), The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer: New York. 

[25]  Kaloshin, D.A. (2001), Search for and stabilization of unstable saddle cycles in the Lorenz system, Differential Equations, 37(11), 16361639. 

[26]  Sarra, S.A. and Meador, C. (2011), On the numerical solution of chaotic dynamical systems using extend precision floating point arithmetic and very high order numerical methods, Nonlinear Analysis: Modelling and Control, 16(3), 340352. 

[27]  Lorenz, E.N., Predictability: does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? American Association for the Advancement of Science, 139th Meeting. AAAS Section on Environmental Sciences New Approaches to Global Weather: GARP (The Global Atmospheric Research Program), December 29, 1972. 

[28]  Motsa, S.S. (2012), A new piecewisequasilinearization method for solving chaotic systems of initial value problems, Central European Journal of Physics, 10(4), 936946. 

[29]  Motsa, S.S., Dlamini, P. and Khumalo, M. (2013), A new multistage spectral relaxation method for solving chaotic initial value systems, Nonlinear Dynamics, 72(1), 265283. 

[30]  Eftekhari, S.A. and Jafari, A.A. (2012), Numerical simulation of chaotic dynamical systems by the method of differential quadrature, Scientia Iranica B, 19(5), 12991315. 

[31]  Chowdhury, M.S.H., Hashim, I. and Momani, S. (2009), The multistage homotopyperturbation method: a powerful scheme for handling the Lorenz system, Chaos, Solitons and Fractals, 40(4), 19291937. 

[32]  Gibbons, A. (1960), A program for the automatic integration of differential equations using the method of Taylor series, The Computer Journal, 3(2), 108111. 

[33]  Rall, L.B. (1981), Automatic Differentiation: Techniques and Applications, SpringerVerlag: Berlin – Heidelberg – New York. 

[34]  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 and Fractals, 28(5), 11491158. 

[35]  Abdulaziz, O., Noor, N.F.M., Hashim, I. and Noorani, M.S.M. (2008), Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos, Solitons and Fractals, 36(5), 14051411. 

[36]  AlSawalha, M.M., Noorani, M.S.M. and Hashim, I. (2009), On accuracy of Adomian decomposition method for hyperchaotic Rössler system, Chaos, Solitons and Fractals, 40(4), 18011807. 

[37]  Pchelintsev, A.N. (2014), Numerical and physical modeling of the dynamics of the Lorenz system, Numerical Analysis and Applications, 7(2), 159167. 

[38]  Lozi, R. and Pchelintsev, A.N. (2015), A new reliable numerical method for computing chaotic solutions of dynamical systems: the Chen attractor case, International Journal of Bifurcation and Chaos, 25(13), 1550187. 

[39]  Lozi, R., Pogonin, V.A. and Pchelintsev, A.N. (2016), A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities, Chaos, Solitons and Fractals, 91, 108114. 

[40]  Liao, S. (2009), On the reliability of computed chaotic solutions of nonlinear differential equations, Tellus,61A, 550564. 

[41]  Liao, S. (2013), On the numerical simulation of propagation of microlevel inherent uncertainty for chaotic dynamic systems, Chaos, Solitons and Fractals, 47, 112. 

[42]  Liao, S. and Wang, P. (2014), On the mathematically reliable longterm simulation of chaotic solutions of Lorenz equation in the interval[0,10000], Science China – Physics, Mechanics & Astronomy, 57(2), 330335. 

[43]  Liao, S. (2014), Physical limit of prediction for chaotic motion of threebody problem, Communications in Nonlinear Science and Numerical Simulation, 19, 601616. 

[44]  Liao, S. and Li, X. (2015), On the inherent selfexcited macroscopic randomness of chaotic threebody systems, International Journal of Bifurcation and Chaos, 29(5), 1530023. 

[45]  Liao, S. (2017), On the clean numerical simulation (CNS) of chaotic dynamic systems, Journal of Hydrodynamics, 29(5), 729747. 

[46]  LlanosPérez, J.A., BetancourtMar, J.A., Cochob, G., Mansilla, R. and NietoVillar, J.M. (2016), Phase transitions in tumor growth: III vascular and metastasis behavior, Physica A: Statistical Mechanics and its Applications, 462, 560568. 

[47]  Arroyo, D., Hernandez, F. and Orúe, A.B. (2017), Cryptanalysis of a classical chaosbased cryptosystem with some quantum cryptography features, International Journal of Bifurcation and Chaos, 27(1), 1750004. 

[48]  Luo, A.C.J. (2015), Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems, International Journal of Bifurcation and Chaos, 25(3), 1550044. 

[49]  Luo, A.C.J. (2015), Discretization and Implicit Mapping Dynamics, Springer: Heidelberg – New York – Dordrecht – London. 

[50]  Wang, P., Liu, Y. and Li, J. (2014), Clean numerical simulation for some chaotic systems using the parallel multipleprecision Taylor scheme, Chinese Science Bulletin, 59(33), 44654472. 

[51]  Mezzarobba, M. and Salvy, B. (2010), Effective bounds for Precursive sequences, Journal of Symbolic Computation, 45(10), 10751096. 

[52]  Chin, P.S.M. (1986), A general method to derive Lyapunov functions for nonlinear systems, International Journal of Control, 44(2), 381393. 

[53]  Leonov, G.A. (2001), Bounds for attractors and the existence of homoclinic orbits in the Lorenz system, Journal of Applied Mathematics and Mechanics, 65(1), 1932. 

[54]  Zhang, F., Shu, Y. and Yang, H. (2011), Bounds for a new chaotic system and its application in chaos synchronization, Communications in Nonlinear Science and Numerical Simulation, 16(3), 15011508. 

[55]  Li, D., Lu, J., Yu, X. and Chen, G. (2005), Estimating the bounds for the Lorenz family of chaotic systems, Chaos, Solitons and Fractals, 23(2), 529534. 

[56]  Wang, P., Li, D. and Hu, Q. (2010), Bounds of the hyperchaotic LorenzStenflo system, Communications in Nonlinear Science and Numerical Simulation, 15(9), 25142520. 

[57]  Wang, P., Li, D., Wu, X., L¨u, J. and Yu, X. (2011), Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems, International Journal of Bifurcation and Chaos, 21(9), 2679. 

[58]  Jafari, S., Sprott, J.C. and Nazarimehr, F. (2015), Recent new examples of hidden attractors, The European Physical Journal Special Topics, 224(8), 14691476. 