On the notion of random chaos

Andres, Jan

doi:10.1090/proc/13464

On the notion of random chaos
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Proc. Amer. Math. Soc. 145 (2017), 3423-3435 Request permission

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.

References

Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, 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, DOI 10.1142/4205
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.
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, DOI 10.1016/j.camwa.2011.11.043
J. Andres, Randomization of Sharkovsky-type results on the circle, Stochastics and Dynamics, doi:10.1142/S0219493717500174.
Jan Andres and PawełBarbarski, Randomized Sharkovsky-type results and random subharmonic solutions of differential inclusions, Proc. Amer. Math. Soc. 144 (2016), no. 5, 1971–1983. MR 3460160, DOI 10.1090/proc/13014
J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003.
Bernd Aulbach and Bernd Kieninger, On three definitions of chaos, Nonlinear Dyn. Syst. Theory 1 (2001), no. 1, 23–37. MR 1989572
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, DOI 10.2307/2324899
Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337–342. MR 431282, DOI 10.1016/0040-9383(76)90026-4
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, DOI 10.1090/S0002-9939-1981-0612745-7
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, DOI 10.1090/S0002-9947-1987-0871677-X
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
R. Graw, On the connection between periodicity and chaos of continuous real functions and their iterates, Aequationes Math. 19 (1979), 277–278.
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
Roman Hric, Topological sequence entropy for maps of the circle, Comment. Math. Univ. Carolin. 41 (2000), no. 1, 53–59. MR 1756926
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, DOI 10.2307/3072372
Abdul Khaleque and Parongama Sen, Effect of randomness in logistic maps, Internat. J. Modern Phys. C 26 (2015), no. 8, 1550086, 11. MR 3342125, DOI 10.1142/S0129183115500862
Milan Kuchta, Characterization of chaos for continuous maps of the circle, Comment. Math. Univ. Carolin. 31 (1990), no. 2, 383–390. MR 1077909
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, DOI 10.1007/978-1-4612-4286-4
Shi Hai Li, $\omega$-chaos and topological entropy, Trans. Amer. Math. Soc. 339 (1993), no. 1, 243–249. MR 1108612, DOI 10.1090/S0002-9947-1993-1108612-8
T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
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.
Megumi Miyazawa, Chaos and entropy for circle maps, Tokyo J. Math. 25 (2002), no. 2, 453–458. MR 1948675, DOI 10.3836/tjm/1244208864
Y. Oono, Period${}\ne 2^n$ implies chaos, Progr. Theor. Phys. 59, 3 (1978), 1028–1030.
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, DOI 10.1090/S0002-9947-2012-05377-3
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.
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, DOI 10.1142/S0218127495000934
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
Jerzy Szulga, Introduction to random chaos, Chapman & Hall, London, 1998. MR 1681553
J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269–282. MR 849479, DOI 10.1090/S0002-9947-1986-0849479-9
Michel Vellekoop and Raoul Berglund, On intervals, transitivity = chaos, Amer. Math. Monthly 101 (1994), no. 4, 353–355. MR 1270961, DOI 10.2307/2975629
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, DOI 10.1103/PhysRevLett.65.2935