Article Contents
Article Contents

# Random interval diffeomorphisms

• We consider a class of step skew product systems of interval diffeomorphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present in a simplified setting intriguing phenomena of intermingled basins, master-slave synchronization and on-off intermittency. We provide a self-contained discussion of these phenomena.

Mathematics Subject Classification: 37E05, 37H10, 37H15.

 Citation:

• Figure 1.  The first frame depicts the graphs of $g_1,g_2$, the diffeomorphisms on $\mathbb{I}$ that are conjugate to the maps $y \mapsto y \pm 1$ that generate the symmetric random walk. The second frame shows a time series of the iterated function system generated by $g_1,g_2$, both picked with probability $1/2$.

Figure 3.  The first frame shows a numerically computed histogram for a time series of the iterated function systems generated by the same diffeomorphisms $f_1^{-1}$ and $f_2^{-1}$ used in Figure 2. The second frame indicates asymptotic convergence of orbits within fibers: it depicts time series for three different initial conditions in $\mathbb{I}$ with the same $\omega$.

Figure 2.  With $r = 1/2$, the diffeomorphisms $f_1 (x) = x - r x (1-x)$ and $f_2 (x) = x + r x (1-x)$ (picked with probabilities $1/2$) give negative Lyapunov exponents at the end points $0,1$. Depicted, in the first frame, are the graphs of the inverse diffeomorphisms $f_1^{-1} (x) = \frac{1 - r - \sqrt{(1-r)^2 +4 r x}}{-2r}$ and $f_2^{-1} (x) = \frac{1 + r - \sqrt{(1+r)^2 -4 r x}}{2r}$. The inverse maps give positive Lyapunov exponents at the end points. The second frame shows a time series for the iterated function system generated by $f_1^{-1}$ and $f_2^{-1}$.

Figure 4.  The first frame depicts the graphs of $x \mapsto f_i(x) = g_i(x) ( 1 - p(x))$, $i=1,2$, with $g_i (x)$ as in (3), (4) and $p(x) = \frac{3}{10} x (1-x)$. The corresponding step skew product system has a zero Lyapunov exponent along $\Sigma_2^+ \times \{0\}$ and a positive Lyapunov exponent along $\Sigma_2^+ \times \{1\}$. The second frame shows a time series for the iterated function system generated by these diffeomorphisms.

Figure 5.  A sequence of stopping times is defined to label subsequent iterates where $y_n$ leaves $(-\infty,K]$ or $(K,\infty)$.

•  J. C. Alexander , J. A. Yorke , Z. You  and  I. Kan , Riddled basins, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992) , 795-813.  doi: 10.1142/S0218127492000446. L. Alsedá  and  M. Misiurewicz , Random interval homeomorphisms, Publ. Mat., 58 (2014) , 15-36. V. A. Antonov , Modeling cyclic evolution processes: Synchronization by means of a random signal, Vestnik Leningrad. Univ. Mat. Mekh. Astronom, 2 (1984) , 67-76. L. Arnold, Random Dynamical Systems Springer Verlag, 1998. doi: 10.1007/978-3-662-12878-7. K. B. Athreya  and  J. Dai , Random logistic maps Ⅰ, Journal of Theoretical Probability, 13 (2000) , 595-608.  doi: 10.1023/A:1007828804691. K. B. Athreya  and  H. J. Schuh , Random logistic maps Ⅱ. The Critical case, Journal of Theoretical Probability, 16 (2003) , 813-830.  doi: 10.1023/B:JOTP.0000011994.90898.81. L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge University Press, 2007. doi: 10.1017/CBO9781107326026. P. H. Baxendale , Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms, Probability Theory and Related Fields, 81 (1989) , 521-554.  doi: 10.1007/BF00367301. P. Bergé, Y. Pomeau and C. Vidal, Order Within Chaos. Towards a Deterministic Approach to Turbulence John Wiley & Sons Ltd. , 1986. C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity Springer-Verlag, 2005. A. Bonifant  and  J. Milnor , Schwarzian derivatives and cylinder maps, Fields Institute Communications, 53 (2008) , 1-21. K. L. Chung, A Course in Probability Theory Harcourt, Brace & World, Inc. 1968. H. Crauel , A uniformly exponential random forward attractor which is not a pullback attractor, Arch. Math., 78 (2002) , 329-336.  doi: 10.1007/s00013-002-8254-9. B. Deroin , V. A. Kleptsyn  and  A. Navas , Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., 199 (2007) , 199-262.  doi: 10.1007/s11511-007-0020-1. M. D. Donsker  and  S. R. S. Varadhan , Asymptotic evaluation of certain Markov process expectations for large time. Ⅰ, Comm. Pure Appl. Math., 28 (1975) , 1-47. J. L. Doob, Measure Theory Springer Verlag, 1994. doi: 10.1007/978-1-4612-0877-8. H. Furstenberg , Boundary theory and stochastic processes on homogeneous spaces, Proc. Sympos. Pure Math., 26 (1973) , 193-229. M. Gharaei  and  A. J. Homburg , Skew products of interval maps over subshifts, J. Difference Equ. Appl., 22 (2016) , 941-958.  doi: 10.1080/10236198.2016.1164146. A. S. Gorodetskiĭ  and  Yu. S. Il'yashenko , Certain new robust properties of invariant sets and attractors of dynamical systems, Funct. Anal. Appl., 33 (1999) , 95-105.  doi: 10.1007/BF02465190. A. S. Gorodetskiĭ , Yu. S. Il'yashenko , V. A. Kleptsyn  and  M. B. Nal'skiĭ , Nonremovability of zero Lyapunov exponents, Funct. Anal. Appl., 39 (2005) , 21-30.  doi: 10.1007/s10688-005-0014-8. J. F. Heagy , N. Platt  and  S. M. Hammel , Characterization of on-off intermittency, Phys. Rev. E., 49 (1994) , 1140-1150.  doi: 10.1103/PhysRevE.49.1140. Yu. S. Il'yashenko , Thick attractors of step skew products, Regular and Chaotic Dynamics, 15 (2010) , 328-334.  doi: 10.1134/S1560354710020188. Yu. S. Il'yashenko , Thick attractors of boundary preserving diffeomorphisms, Indag. Math. (N.S.), 22 (2011) , 257-314.  doi: 10.1016/j.indag.2011.09.006. Yu. S. Il'yashenko , V. A. Kleptsyn  and  P. Saltykov , Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins, J. Fixed Point Theory Appl., 3 (2008) , 449-463.  doi: 10.1007/s11784-008-0088-z. I. Kan , Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc., 31 (1994) , 68-74.  doi: 10.1090/S0273-0979-1994-00507-5. G. Keller , Stability index for chaotically driven concave maps, J. London Math. Soc., 89 (2014) , 603-622.  doi: 10.1112/jlms/jdt070. V. A. Kleptsyn  and  M. B. Nal'skiĭ , Contraction of orbits in random dynamical systems on the circle, Funct. Anal. Appl., 38 (2004) , 267-282.  doi: 10.1007/s10688-005-0005-9. V. A. Kleptsyn  and  P. S. Saltykov , On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps, Trans. Moscow Math. Soc., 72 (2011) , 193-217.  doi: 10.1090/s0077-1554-2012-00196-4. V. A. Kleptsyn  and  D. Volk , Physical measures for nonlinear random walks on interval, Mosc. Math. J., 14 (2014) , 339-365. P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems Amer. Math. Soc. , 2011. doi: 10.1090/surv/176. J. Lamperti , Criteria for the recurrence or transience of stochastic processes, Ⅰ, J. Math. Anal. and Appl., 1 (1960) , 314-330.  doi: 10.1016/0022-247X(60)90005-6. J. Lamperti , A new class of probability limit theorems, J. Math. and Mech., 11 (1962) , 749-772. J. Lamperti , Criteria for stochastic processes Ⅱ: Passage-time moments, J. Math. Anal. and Appl., 7 (1963) , 127-145.  doi: 10.1016/0022-247X(63)90083-0. J. Milnor , On the concept of attractor, Commun. Math. Phys., 99 (1985) , 177-195.  doi: 10.1007/BF01212280. E. Ott , J. Sommerer , J. Alexander , I. Kan  and  J. Yorke , Scaling behavior of chaotic systems with riddled basins, Physical Review Letters, 71 (1993) , 4134-4137.  doi: 10.1103/PhysRevLett.71.4134. L. M. Pecora  and  T. L. Carroll , Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990) , 821-824.  doi: 10.1103/PhysRevLett.64.821. A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences Cambridge University Press, 2001. doi: 10.1017/CBO9780511755743. N. Platt , E. A. Spiegel  and  C. Tresser , On-off intermittency: A mechanism for bursting, Phys. Rev. Letters, 70 (1993) , 279-282.  doi: 10.1103/PhysRevLett.70.279. A. N. Shiryayev, Probability Springer Verlag, 1984. doi: 10.1007/978-1-4899-0018-0. J. Stark , Invariant graphs for forced systems, Phys. D, 109 (1997) , 163-179.  doi: 10.1016/S0167-2789(97)00167-X. M. Viana, Lectures on Lyapunov Exponents Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602. H. Zmarrou  and  A. J. Homburg , Bifurcations of stationary densities of random diffeomorphisms, Ergod. Th. Dyn. Systems, 27 (2007) , 1651-1692.  doi: 10.1017/S0143385707000077. H. Zmarrou  and  A. J. Homburg , Dynamics and bifurcations of random circle diffeomorphisms, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008) , 719-731.  doi: 10.3934/dcdsb.2008.10.719.

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