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Article

  • Title

    COMPUTATIONAL ASPECTS OF LARGE-LENGTH CYCLE SEARCH ALGORITHMS FOR NONLINEAR DISCRETE SYSTEMS

  • Authors

    Skrynnyk I.
    Dmitrishin D. V.
    Stokolos A.
    Якоб Іонут Еміл

  • Subject

    COMPUTER AND INFORMATION NETWORKS AND SYSTEMS. MANUFACTURING AUTOMATION

  • Year 2019
    Issue 2(58)
    UDC 517.9/519.6
    DOI 10.15276/opu.2.58.2019.08
    Pages 69-84
  • Abstract

    Even the simplest nonlinear discrete systems dynamics is very complex. It includes both periodic movements and quasi-periodic or recurrent ones. In such systems, almost always present are the chaotic attractors, whose nature is currently well studied, at least for a wide class of model equations. In many cases, chaotic attractors can be modeled using periodic motions characterized with large periods. Such attractors’ and minimal invariant sets’ search represents an important task of applied mathematics, with respect to that the solutions are used in physical, chemical, economic sciences, in coding theory, signal transmission theory and so on. However, mathematical results based on computer calculations require a careful verification, since these calculations themselves are carried out approximately, and the chaotic systems are very sensitive to calculation errors. One of the approaches to solving the cycles search and verification problem is based on the application of these cycles’ stabilization methods. These methods can be divided into two groups: delayed control, that uses knowledge on system’s previous states, and predictive control, which uses the future values of system state in the absence of control. This study purpose is to demonstrate the effectiveness of the cycles search averaged predictive control method on some dynamical systems widely referred to in technical reference sources. Another important goal we aimed onto is to formulate the necessary conditions at which the orbit found actually represents a cycle. The article exposes the elaboration of predictive control methods: the averaged predictive control is used, at that the cycles search algorithms based on such control properties are offered. Noted are various features of algorithms’ functioning that depend on the original discrete system properties. Proposed are the cyclic points’ verification methods in the form of three necessary conditions of point’s cyclicity: checking the smallness of the residual, checking the periodicity and checking the cycle local asymptotic stability. Well-known two-dimensional discrete systems such as Lozi, Henon, Ikeda, Elhadj-Sprott, Multihorseshoe, Prey-Predator have been chosen to demonstrate the algorithm and numerical simulation. These systems’ essential features include the presence of large lengths cycles with a dominant multiplier, i.e. when two-dimensional case one multiplier has larger modulus, and another’s modulus is less than one. With this class of systems, the proposed algorithm operates particularly efficiently. The developed method can also be used to study the discrete dynamical systems’ topological properties dependence on changes in parameters, as well as to study the presence of bifurcations and their types.

  • Keywords nonlinear discrete systems, periodic solutions stabilization, search algorithms for large-length cycles
  • Viewed: 170 Dowloaded: 1
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  • References

    Література

     

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    15. Dmitrishin D., Khamitova A. Methods of harmonic analysis in nonlinear dynamics. Comptes Rendus Mathematique. 2013. Vol. 351, Is. 9–10. P. 367–370.

    16. Morgul O. Further stability results for a generalization of delayed feedback control. Nonlinear Dynam-ics. 2012. 70 (2). P. 1–8. DOI: 10.1007/s11071-012-0530-z.

    17. Polyak B.T. Stabilizing chaos with predictive control. Automation and Remote Control. 2005. 66(11). P. 1791–1804.

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    19. Dmitrishin D., Hagelstein P., Khamitova A., Stokolos A. Limitations of Robust Stability of a Linear Delayed Feedback Control. SIAM J. Control Optim. 2018. 56. P. 148–157.

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    26. H ́enon M. A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 1976. 50(1). P. 69–77.

    27. Ikeda K. Multiple-valued Stationary State and its Instability of the Transmitted Light by a Ring Cavity System. Opt. Commun. 1979. 30. P. 257–261.

    28. Elhadj Z., Sprott J.C. A two-dimensional discrete mapping with С multifold chaotic attractors. Elec. J. Theoretical Phys. 2008. 5(17). P. 1–14. URL: http://thor.physics.wisc.edu/chaos/elhadj/2dmap3/2dmap3.pdf.

    29. Joshi Y., Blackmore D. Strange Attractors for Asymptotically Zero Maps. Chaos, Solitons & Fractals. 2014. Vol. 68. P. 123–138. DOI: https://doi.org/10.1016/j.chaos.2014.08.005.

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    31. Sauer T. Computer arithmetic and sensitivity of natural measure. Journal of Difference Equations and Applications. 2005. 11, No.7. P. 669–676. DOI: 10.1080/10236190412331334545.

    32. Dmitrishin D., SkrinnikI., Lesaja G., Stokolos A. A new method for finding cycles by semilinear con-trol. Physics Letters A. 2019. 383. P. 1871–1878. DOI: https://doi.org/10.1016/j.physleta.2019.03.013.

    33. Shalby L. Predictive feedback control method for stabilization of continuous time systems. Advances in Systems Science and Applications. 2017. 17. P. 1–13.

     

    References

     

    1. Dmitrishin, D., Stokolos, A., & Iacob, E. (2019). Average predictive control for nonlinear discrete dy-namical systems. arXiv:1906.02925. Retrieved from: https://arxiv.org/abs/1906.02925.

    2. Devaney, R.L. (1993). An Introduction to Chaotic Dynamical Systems. New York: Addison-Wesley Publ. Co., Second Edition.

    3. Ott, E., Grebodgi, C., & Yorke, J.A. (1990). Controlling chaos. Phys. Rev. Lett. 64, 1196-1199. DOI: https://doi.org/10.1103/PhysRevLett.64.1196.

    4. Chen, G., & Dong, X. (1999). From chaos to order: Methodologies, Perspectives and Application. Sin-gapore: World Scientific.

    5. Zygliczy´nski, P. (1997). Computer assisted proof of chaos in the Rössler equations and the Hénon map. Nonlinearity, 10 (1), 243–252.

    6. Galias, Z. (2002). Rigorous investigations of Ikeda map by means of interval arithmetic. Nonlinearity, 15, 1759–1779.

    7. Galias, Z. (2001). Interval methods for rigorous investigations of periodic orbits. Int. J. Bifurc. Chaos, 11 (9), 2427–2450.

    8. Lozi, R. (2013). Can we trust in numerical computations of chaotic solutions of dynamical systems? Topology and Dynamics of Chaos, World Scientific Series in Nonlinear Science Series A, 84, 63–98. DOI: https://doi.org/10.1142/9789814434867_0004.

    9. Yang, D., & Zhou, J. (2014). Connections among several chaos feedback control approaches and cha-otic vibration control of mechanical systems. Commun. Nonlinear Sci. Numer. Simulat, 19, 3954–3968. DOI: 10.1016/j.cnsns.2014.04.001.

    10. Andrievsky, B.R., & Fradkov, A.L. (2003). Control of Chaos: Methods and Applications. I. Methods. Avtomatikai Telemekhanika, 5, 3–45.

    11. Miller, J.R., & Yorke, J.A. (2000). Finding all periodic orbits of maps using Newton methods: sizes of basins.Physica D, 135, 195–211. DOI: https://doi.org/10.1016/S0167-2789(99)00138-4.

    12. Ypma, T.J. (1995). Historical Development of the Newton-Raphson Method. SIAM Rev., 37(4), 531–551. DOI: https://doi.org/10.1137/1037125.

    13. Pyragas, K. (1992). Continuous control of chaos by self controlling feedback. Phys. Rev. Lett. A, 170 (6), 421–428. DOI: https://doi.org/10.1016/0375-9601(92)90745-8.

    14. Vieira de S.M., & Lichtenberg, A.J. (1996). Controlling chaos using nonlinear feedback with delay. Phys. Rev. E, 54, 1200–1207. DOI: https://doi.org/10.1103/PhysRevE.54.1200.

    15. Dmitrishin, D., & Khamitova, A. (2013). Methods of harmonic analysis in nonlinear dynamics. Comptes Rendus Mathematique, 351, 9–10, 367–370.

    16. Morgul, O. (2012). Further stability results for a generalization of delayed feedback control. Nonlinear Dynamics, 70 (2), 1–8. DOI: 10.1007/s11071-012-0530-z.

    17. Polyak, B.T. (2005). Stabilizing chaos with predictive control. Automation and Remote Control, 66(11), 1791–1804.

    18. Ushio, T., Yamamoto, S. (1999). Prediction-based control of chaos. Phys. Lett. A, 264, 30–35.

    19. Dmitrishin, D., Hagelstein, P., Khamitova, A., & Stokolos, A. (2018). Limitations of Robust Stability of a Linear Delayed Feedback Control. SIAM J. Control Optim. 56, 148–157.

    20. Zhu, J., & Tian, Y-P. (2005). Necessary and sufficient condition for stabilizability of discrete-time sys-tems via delayed feedback control. Phys. Lett. A, 343, 95–107.

    21. Ushio, T. (1996). Limitation of delayed feedback control in nonlinear discrete-time systems. IEEE Trans. Circuits Syst., 43 (9), 815–816.

    22. Sakai, K. (1996). Diffeomorphisms with the shadowing property. J. Austral. Math. Soc. (Series A), 1, 396–399.

    23. Kuntsevich, A.V., & Kuntsevich, V.M. (2012). Estimates of Stable Limit Cycles of Nonlinear Discrete Systems. Journal of Automation and Information Sciences, 44, 9, 1–10.

    24. Sprott, J. C. (2003). Chaos and Time-Series Analysis. Oxford University Press, Oxford, UK, & New York.

    25. Lozi, R. (1978). Un attracteur ́etrange du type attracteur de H ́enon. Journal de Physique, 39, 5–9.

    26. H ́enon, M. (1976). A two-dimensional mapping with a strange attractor. Commun. Math. Phys., 50(1), 69–77.

    27. Ikeda, K. (1979). Multiple-valued Stationary State and its Instability of the Transmitted Light by a Ring Cavity System. Opt. Commun, 30, 257–261.

    28. Elhadj, Z., & Sprott, J. C. (2008). A two-dimensional discrete mapping with С multifold chaotic attrac-tors. Elec. J. Theoretical Phys., 5 (17), 1–14. Retrieved from: http://thor.physics.wisc.edu/chaos/elhadj/ 2dmap3/2dmap3.pdf.

    29. Joshi, Y., & Blackmore, D. (2014). Strange Attractors for Asymptotically Zero Maps. Chaos, Solitons & Fractals, 68, 123–138. DOI: https://doi.org/10.1016/j.chaos.2014.08.005.

    30. Beddington, J.R., Free, C.A., & Lawton, J.H. (1975). Dynamic complexity in predator-prey models framed in difference equations. Nature, 225, 58–60. DOI: 10.1038/255058a0.

    31. Sauer, T. (2005). Computer arithmetic and sensitivity of natural measure. Journal of Difference Equa-tions and Applications, 11, 7, 669–676. DOI: 10.1080/10236190412331334545.

    32. Dmitrishin, D., Skrinnik, I., Lesaja, G., & Stokolos, A. (2019). A new method for finding cycles bysemilinear control. Physics Letters A, 383, 1871–1878. DOI: https://doi.org/10.1016/j.physleta.2019.03.013.

    33. Shalby, L. (2017). Predictive feedback control method for stabilization of continuous time systems. Ad-vances in Systems Science and Applications, 17, 1–13.

     

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