Modular Chaos for Random Processes

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Authors

  • Marat Akhmet Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey Author

DOI:

https://doi.org/10.5890/DNC.2022.06.001

Abstract

We apply the method of domain structured chaos for a dynamics with orbits whirling among infinitely many modules. Thus, the conservative conditions of chaos existence are weakened to model irregular dynamics. The research proves that the Poincar\'{e} chaos is of exceptional use for analysis of stochastic processes. Examples, illustrating the results are provided.

References

[1] Devaney, R.L. (1987), An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Menlo Park.

[2] Luo, A. (2017), Periodic Flows to Chaos in Time-delay Systems, Springer, New York.

[3] Li, T.Y. and Yorke, J.A. (1975), Period Three Implies Chaos, The American Mathematical Monthly, 82, 985-992.

[4] Akhmet, M. and Fen, M.O. (2016), Unpredictable points and chaos, Communications in Nonlinear Science and Numerical Simulation, 40, 1-5.

[5] Akhmet, M. and Fen, M.O. (2016), Poincar {e} chaos and unpredictable functions, Communications in Nonlinear Science and Numerical Simulation, 48, 85-94.

[6] Akhmet, M. (2019), Domain-structured chaos in discrete random processes, Arxiv e-prints, arXiv:1912.10478 (submitted).

[7] Akhmet, M. and Alejaily, E.M. (2019), Domain-Structured Chaos in a Hopfield Neural Network, International Journal of Bifurcation and Chaos, 29(14), Article number 1950205.

[8] Akhmet, M. and Alejaily, E.M. (2020), Abstract similarity, chaos and fractals, Discrete and Continuous Dynamical Systems Ser. B., 22, number 1531-3492_2017_11_638, DOI:10.3934/dcdsb.2020191.

[9] Akhmet, M. and Alejaily, E.M. (2021), Abstract fractals, Discontinuity, Nonlinearity and Complexity, 10(1), 135-142.

[10] Akhmet, M., Fen, M.O., and Alejaily, E.M. (2021), Dynamics with fractals, Discontinuity, Nonlinearity and Complexity, 10(1), 135-142.

[11] Akhmet, M., Fen, M.O., and Alejaily, E. M. (2020), Dynamics with chaos and fractals, Springer.

[12] Akhmet, M. (2022) Abstract hyperbolic chaos, Discontinuity, Nonlinearity and Complexity, 11(1), 133-138.

[13] Zmeskal, O., Dzik, P., and Vesely, M. (2013), Entropy of fractal systems, Computers and Mathematics with Applications, 66(2), 135-146.

[14] Moran, P.A.P. (1946), Additive functions of intervals and Hausdorff measure, Proceedings of the Cambridge Philosophical Society, 42, 15-23.

[15] Jorgensen, P.E.T. (2006), Analysis and Probability: Wavelets, Signals, Fractals, Graduate Texts in Mathematics, 234, Springer, New York.

[16] Stella, S. (1992), On Hausdorff Dimension of Recurrent Net Fractals, Proceedings of the American Mathematical Society, 116, 389-400.

[17] Pesin, Y. and Weiss, H. (1996), On the Dimension of Deterministic and Random Cantor-like Sets, Symbolic Dynamics, and the Eckmann-Ruelle Conjecture, Communications In Mathematical Physics, 182, 105-153.

[18] Hutchinson, J. (1981), Fractals and self-similarity, Indiana University Mathematics Journal, 30(5), 713-747.

[19] Hata, M. (1985), On the structure of self-similar sets, Japan J. Applied Mathematics, 2, 381-414.

[20] Edgar, G.A. (1990), Measure, Topology, and Fractal Geometry, Springer-Verlag, New York.

[21] Spear, D.W. (1992), Measure and self-similarity, Advances in Mathematics, 91, 143-157.

[22] Bandt, C. and Graf, S. (1992), Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure, Proceedings of the American Mathematical Society, 114, 995-1001.

[23] Falconer, K.J. (1995), Sub-self-similar sets, Transactions of the American Mathematical Society, 347(4), 3121-3129.

[24] Lau, K.S., Ngai, S.M., and Rao, H. (2001), Iterated function systems with overlaps and the self-similar measures, Journal of the London Mathematical Society, 63(01), 99-115.

[25] Ngai, S.M. and Wang, Y. (2001), Hausdorff dimension of overlapping self-similar sets, Journal of the London Mathematical Society, 63(2), 655-672.

[26] Wiggins, S. (1988), Global Bifurcation and Chaos: Analytical Methods, Springer-Verlag, New York, Berlin.

[27] Gonz{a}les-Miranda, J.M. (2004), Synchronization and control of chaos, Imperial College Press, London.

[28] Oksendal, B.K. (2003), Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin.

[29] Gagniuc, P.A. (2017), Markov Chains: From Theory to Implementation and Experimentation, USA, NJ: John Wiley and Sons.

[30] Akhmet, M. and Fen, M.O. (2016), Replication of chaos in neural networks, economics and physics, Nonlinear Physical Science Book Series, Springer, Higher Education Press, Beijing, Heidelberg.

[31] Akhmet, M. (2020), Almost periodicity, chaos and asymptotic equivalence, Springer.

[32] Akhmet, M. (2009), Devaney's chaos of a relay system, Commun. Nonlinear Sci. Numer. Simulat., 14, 1486-1493.

[33] Akhmet, M. (2009), Li-Yorke chaos in the system with impacts, J. Math. Anal. Appl., 351, 804-810.

[34] Akhmet, M., Akhmetova, Z., and Fen, M.O. (2016), Exogenous versus endogenous for chaotic business cycles, Discontinuity, Nonlinearity, and Complexity, 5(2), 101-119.

[35] Akhmet, M., Fen, M.O. and Kashkynbayev, A. (2017), Persistence of Li-Yorke chaos in systems with relay, Electron. J. Qual. Theory Differ. Equ., 72, 1-18.

[36] Akhmet, M., Feckan, M., Fen, M.O., and Kashkynbayev, A. (2018), Perturbed Li-Yorke homoclinic chaos, Electronic Journal of Qualitative Theory of Differential Equations, 75, 1-18.

[37] Akhmet, M., Deterministic chaos for Markov chains, Arxiv e-prints, arXiv:2010.14625 (submitted).

[38] Lasota, A. and Mackey, M.C. (1985), Probabilistic Properties of Deterministic Systems, Cambridge, Cambridge Univ. Press.

[39] Ornstein, D. (2004), Kolmogorov, random processes, and Newtonian dynamics, Russian Mathematical Surveys, 59(2), 313-317.

[40] Ornstein, D. (1995), In what Sense can a Deterministic System be Random?, Chaos, solitons and fractals, 5(2), 139-141.

[41] Eisencraft, M., Monteiro, L.H.A., and Soriano, D.C. (2017), White Gaussian Chaos, IEEE communications letters, 21(8), 17-19.

[42] Chen, G. and Huang, Y. (2011), Chaotic Maps: Dynamics, Fractals and Rapid Fluctuations, Synthesis Lectures on Mathematics and Statistics, Morgan and Claypool Publishers, Texas.

[43] Doob, J.L. (1953), Stochastic processes, Wiley, New York.

[44] Sell, G. (1971), Topological dynamics and ordinary differential equations, Van Nostrand Reinhold mathematical studies, 33, New York.

[45] Nicolis, G. and Prigogine, I. (1989), Exploring complexity: An Introduction, Freeman and company, New York.

[46] Akhmet, M. and Tola, A. (2020), Unpredictable Strings, Kazakh Math. J., 20{3}, 16-22.

[47] Williams, S. (editor), (2004), Symbolic Dynamics and its Applications, Proceedings of Symposia in Applied Mathematics 60.

[48] Adler, R., Konheim, A., and McAndrew, M. (1965), Topological entropy, Transactions of the American Mathematical Society, 114 309-319.

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PublishedJune 2022

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How to Cite

Akhmet, M. (2026). Modular Chaos for Random Processes. Discontinuity, Nonlinearity, and Complexity, 11(2), 191-201. https://doi.org/10.5890/DNC.2022.06.001