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On the notion of random chaos


Author: Jan Andres
Journal: Proc. Amer. Math. Soc. 145 (2017), 3423-3435
MSC (2010): Primary 37D45, 37E15; Secondary 37H10
DOI: https://doi.org/10.1090/proc/13464
Published electronically: January 25, 2017
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Abstract: Deterministic chaos is investigated for random dynamical systems in dimension one. Some well-known as well as new Li-Yorke-type theorems are randomized. Deterministic chaos exhibited by random dynamics is therefore called random chaos for brevity. Chaotic random dynamics are also studied for multivalued maps.


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  • [1] Lluís Alsedà, Jaume Llibre, and Michał Misiurewicz, Combinatorial dynamics and entropy in dimension one, 2nd ed., Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1807264
  • [2] J. Andres, Randomization of Sharkovskii-type theorems, Proc. Amer. Math. Soc. 136, 4 (2008), 1385-1395; Erratum: Proc. Amer. Math. Soc. 136, 10 (2008), 3733-3734.
  • [3] Jan Andres, Period two implies chaos for a class of multivalued maps: a naive approach, Comput. Math. Appl. 64 (2012), no. 7, 2160-2165. MR 2966851, https://doi.org/10.1016/j.camwa.2011.11.043
  • [4] J. Andres, Randomization of Sharkovsky-type results on the circle, Stochastics and Dynamics, doi:10.1142/S0219493717500174.
  • [5] Jan Andres and Paweł Barbarski, Randomized Sharkovsky-type results and random subharmonic solutions of differential inclusions, Proc. Amer. Math. Soc. 144 (2016), no. 5, 1971-1983. MR 3460160, https://doi.org/10.1090/proc/13014
  • [6] J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003.
  • [7] Bernd Aulbach and Bernd Kieninger, On three definitions of chaos, Nonlinear Dyn. Syst. Theory 1 (2001), no. 1, 23-37. MR 1989572
  • [8] J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly 99 (1992), no. 4, 332-334. MR 1157223, https://doi.org/10.2307/2324899
  • [9] Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337-342. MR 0431282
  • [10] Louis Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc. 82 (1981), no. 3, 481-486. MR 612745, https://doi.org/10.2307/2043966
  • [11] Louis Block and Ethan M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300 (1987), no. 1, 297-306. MR 871677, https://doi.org/10.2307/2000600
  • [12] Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376
  • [13] R. Graw, On the connection between periodicity and chaos of continuous real functions and their iterates, Aequationes Math. 19 (1979), 277-278.
  • [14] M. C. Hidalgo, Periods of periodic points for transitive degree one maps of the circle with a fixed point, Acta Math. Univ. Comenian. (N.S.) 61 (1992), no. 1, 11-19. MR 1205855
  • [15] Roman Hric, Topological sequence entropy for maps of the circle, Comment. Math. Univ. Carolin. 41 (2000), no. 1, 53-59. MR 1756926
  • [16] Chun-Hung Hsu and Ming-Chia Li, Transitivity implies period six: a simple proof, Amer. Math. Monthly 109 (2002), no. 9, 840-843. MR 1933705, https://doi.org/10.2307/3072372
  • [17] Abdul Khaleque and Parongama Sen, Effect of randomness in logistic maps, Internat. J. Modern Phys. C 26 (2015), no. 8, 1550086, 11. MR 3342125, https://doi.org/10.1142/S0129183115500862
  • [18] Milan Kuchta, Characterization of chaos for continuous maps of the circle, Comment. Math. Univ. Carolin. 31 (1990), no. 2, 383-390. MR 1077909
  • [19] Andrzej Lasota and Michael C. Mackey, Chaos, fractals, and noise, 2nd ed., Applied Mathematical Sciences, vol. 97, Springer-Verlag, New York, 1994. Stochastic aspects of dynamics. MR 1244104
  • [20] Shi Hai Li, $ \omega $-chaos and topological entropy, Trans. Amer. Math. Soc. 339 (1993), no. 1, 243-249. MR 1108612, https://doi.org/10.2307/2154217
  • [21] Tien Yien Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985-992. MR 0385028
  • [22] V. Loreto, G. Paladin, M. Pasquini, and A. Vulpiani, Characterization of chaos in random maps, Physica A: Statistical Mechanics and its Applications 232, 1-2 (1996), 189-200.
  • [23] Megumi Miyazawa, Chaos and entropy for circle maps, Tokyo J. Math. 25 (2002), no. 2, 453-458. MR 1948675, https://doi.org/10.3836/tjm/1244208864
  • [24] Y. Oono, Period$ {}\ne 2^n$ implies chaos, Progr. Theor. Phys. 59, 3 (1978), 1028-1030.
  • [25] Brian E. Raines and David R. Stockman, Fixed points imply chaos for a class of differential inclusions that arise in economic models, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2479-2492. MR 2888216, https://doi.org/10.1090/S0002-9947-2012-05377-3
  • [26] S. Rim, D.-U. Hwang, I. Kim, and C.-M. Kim, Chaotic transition of random dynamics systems and chaos synchronization by common noises, Phys. Rev. Lett. 85, 11 (2000), 2304-2307.
  • [27] A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Proceedings of the Conference ``Thirty Years after Sharkovskiĭ's Theorem: New Perspectives'' (Murcia, 1994), 1995, pp. 1263-1273. Translated from the Russian [Ukrain. Mat. Zh. 16 (1964), no. 1, 61-71; MR0159905 (28 #3121)] by J. Tolosa. MR 1361914, https://doi.org/10.1142/S0218127495000934
  • [28] H. W. Siegberg, Chaotic mappings on $ S^{1}$, periods one, two, three imply chaos on $ S^{1}$, Numerical solution of nonlinear equations (Bremen, 1980) Lecture Notes in Math., vol. 878, Springer, Berlin-New York, 1981, pp. 351-370. MR 644337
  • [29] Jerzy Szulga, Introduction to random chaos, Chapman & Hall, London, 1998. MR 1681553
  • [30] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269-282. MR 849479, https://doi.org/10.2307/2000468
  • [31] Michel Vellekoop and Raoul Berglund, On intervals, transitivity = chaos, Amer. Math. Monthly 101 (1994), no. 4, 353-355. MR 1270961, https://doi.org/10.2307/2975629
  • [32] Lei Yu, Edward Ott, and Qi Chen, Transition to chaos for random dynamical systems, Phys. Rev. Lett. 65 (1990), no. 24, 2935-2938. MR 1081067, https://doi.org/10.1103/PhysRevLett.65.2935

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Additional Information

Jan Andres
Affiliation: Department of Mathematical Analysis and Application of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
Email: jan.andres@upol.cz

DOI: https://doi.org/10.1090/proc/13464
Keywords: Random chaos, deterministic chaos, randomization, chaotic random dynamics, random periodic orbits, multivalued maps
Received by editor(s): February 11, 2016
Received by editor(s) in revised form: September 6, 2016
Published electronically: January 25, 2017
Additional Notes: The author was supported by the grant No. 14-06958S “Singularities and impulses in boundary value problems for nonlinear ordinary differential equations” of the Grant Agency of the Czech Republic.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

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