April  2017, 10(2): 241-272. doi: 10.3934/dcdss.2017012

Random interval diffeomorphisms

1. 

KdV Institute for Mathematics, University of Amsterdam, Science park 107,1098 XG Amsterdam, Netherlands

2. 

Department of Mathematics, VU University Amsterdam, De Boelelaan 1081,1081 HV Amsterdam, Netherlands

Received  October 2015 Revised  November 2016 Published  January 2017

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.

Citation: Masoumeh Gharaei, Ale Jan Homburg. Random interval diffeomorphisms. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 241-272. doi: 10.3934/dcdss.2017012
References:
[1]

J. C. AlexanderJ. A. YorkeZ. You and I. Kan, Riddled basins, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 795-813.  doi: 10.1142/S0218127492000446.  Google Scholar

[2]

L. Alsedá and M. Misiurewicz, Random interval homeomorphisms, Publ. Mat., 58 (2014), 15-36.   Google Scholar

[3]

V. A. Antonov, Modeling cyclic evolution processes: Synchronization by means of a random signal, Vestnik Leningrad. Univ. Mat. Mekh. Astronom, 2 (1984), 67-76.   Google Scholar

[4]

L. Arnold, Random Dynamical Systems Springer Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[5]

K. B. Athreya and J. Dai, Random logistic maps Ⅰ, Journal of Theoretical Probability, 13 (2000), 595-608.  doi: 10.1023/A:1007828804691.  Google Scholar

[6]

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.  Google Scholar

[7]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge University Press, 2007. doi: 10.1017/CBO9781107326026.  Google Scholar

[8]

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.  Google Scholar

[9]

P. Bergé, Y. Pomeau and C. Vidal, Order Within Chaos. Towards a Deterministic Approach to Turbulence John Wiley & Sons Ltd. , 1986.  Google Scholar

[10]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity Springer-Verlag, 2005.  Google Scholar

[11]

A. Bonifant and J. Milnor, Schwarzian derivatives and cylinder maps, Fields Institute Communications, 53 (2008), 1-21.   Google Scholar

[12]

K. L. Chung, A Course in Probability Theory Harcourt, Brace & World, Inc. 1968.  Google Scholar

[13]

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.  Google Scholar

[14]

B. DeroinV. 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.  Google Scholar

[15]

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.   Google Scholar

[16]

J. L. Doob, Measure Theory Springer Verlag, 1994. doi: 10.1007/978-1-4612-0877-8.  Google Scholar

[17]

H. Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Proc. Sympos. Pure Math., 26 (1973), 193-229.   Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

A. S. GorodetskiĭYu. S. Il'yashenkoV. 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.  Google Scholar

[21]

J. F. HeagyN. Platt and S. M. Hammel, Characterization of on-off intermittency, Phys. Rev. E., 49 (1994), 1140-1150.  doi: 10.1103/PhysRevE.49.1140.  Google Scholar

[22]

Yu. S. Il'yashenko, Thick attractors of step skew products, Regular and Chaotic Dynamics, 15 (2010), 328-334.  doi: 10.1134/S1560354710020188.  Google Scholar

[23]

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.  Google Scholar

[24]

Yu. S. Il'yashenkoV. 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.  Google Scholar

[25]

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.  Google Scholar

[26]

G. Keller, Stability index for chaotically driven concave maps, J. London Math. Soc., 89 (2014), 603-622.  doi: 10.1112/jlms/jdt070.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

V. A. Kleptsyn and D. Volk, Physical measures for nonlinear random walks on interval, Mosc. Math. J., 14 (2014), 339-365.   Google Scholar

[30]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems Amer. Math. Soc. , 2011. doi: 10.1090/surv/176.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Lamperti, A new class of probability limit theorems, J. Math. and Mech., 11 (1962), 749-772.   Google Scholar

[33]

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.  Google Scholar

[34]

J. Milnor, On the concept of attractor, Commun. Math. Phys., 99 (1985), 177-195.  doi: 10.1007/BF01212280.  Google Scholar

[35]

E. OttJ. SommererJ. AlexanderI. 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.  Google Scholar

[36]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar

[37]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences Cambridge University Press, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar

[38]

N. PlattE. 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.  Google Scholar

[39]

A. N. Shiryayev, Probability Springer Verlag, 1984. doi: 10.1007/978-1-4899-0018-0.  Google Scholar

[40]

J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179.  doi: 10.1016/S0167-2789(97)00167-X.  Google Scholar

[41]

M. Viana, Lectures on Lyapunov Exponents Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.  Google Scholar

[42]

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.  Google Scholar

[43]

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.  Google Scholar

show all references

References:
[1]

J. C. AlexanderJ. A. YorkeZ. You and I. Kan, Riddled basins, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 795-813.  doi: 10.1142/S0218127492000446.  Google Scholar

[2]

L. Alsedá and M. Misiurewicz, Random interval homeomorphisms, Publ. Mat., 58 (2014), 15-36.   Google Scholar

[3]

V. A. Antonov, Modeling cyclic evolution processes: Synchronization by means of a random signal, Vestnik Leningrad. Univ. Mat. Mekh. Astronom, 2 (1984), 67-76.   Google Scholar

[4]

L. Arnold, Random Dynamical Systems Springer Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[5]

K. B. Athreya and J. Dai, Random logistic maps Ⅰ, Journal of Theoretical Probability, 13 (2000), 595-608.  doi: 10.1023/A:1007828804691.  Google Scholar

[6]

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.  Google Scholar

[7]

L. Barreira and Y. Pesin, Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents, Cambridge University Press, 2007. doi: 10.1017/CBO9781107326026.  Google Scholar

[8]

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.  Google Scholar

[9]

P. Bergé, Y. Pomeau and C. Vidal, Order Within Chaos. Towards a Deterministic Approach to Turbulence John Wiley & Sons Ltd. , 1986.  Google Scholar

[10]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity Springer-Verlag, 2005.  Google Scholar

[11]

A. Bonifant and J. Milnor, Schwarzian derivatives and cylinder maps, Fields Institute Communications, 53 (2008), 1-21.   Google Scholar

[12]

K. L. Chung, A Course in Probability Theory Harcourt, Brace & World, Inc. 1968.  Google Scholar

[13]

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.  Google Scholar

[14]

B. DeroinV. 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.  Google Scholar

[15]

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.   Google Scholar

[16]

J. L. Doob, Measure Theory Springer Verlag, 1994. doi: 10.1007/978-1-4612-0877-8.  Google Scholar

[17]

H. Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Proc. Sympos. Pure Math., 26 (1973), 193-229.   Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

A. S. GorodetskiĭYu. S. Il'yashenkoV. 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.  Google Scholar

[21]

J. F. HeagyN. Platt and S. M. Hammel, Characterization of on-off intermittency, Phys. Rev. E., 49 (1994), 1140-1150.  doi: 10.1103/PhysRevE.49.1140.  Google Scholar

[22]

Yu. S. Il'yashenko, Thick attractors of step skew products, Regular and Chaotic Dynamics, 15 (2010), 328-334.  doi: 10.1134/S1560354710020188.  Google Scholar

[23]

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.  Google Scholar

[24]

Yu. S. Il'yashenkoV. 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.  Google Scholar

[25]

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.  Google Scholar

[26]

G. Keller, Stability index for chaotically driven concave maps, J. London Math. Soc., 89 (2014), 603-622.  doi: 10.1112/jlms/jdt070.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

V. A. Kleptsyn and D. Volk, Physical measures for nonlinear random walks on interval, Mosc. Math. J., 14 (2014), 339-365.   Google Scholar

[30]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems Amer. Math. Soc. , 2011. doi: 10.1090/surv/176.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Lamperti, A new class of probability limit theorems, J. Math. and Mech., 11 (1962), 749-772.   Google Scholar

[33]

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.  Google Scholar

[34]

J. Milnor, On the concept of attractor, Commun. Math. Phys., 99 (1985), 177-195.  doi: 10.1007/BF01212280.  Google Scholar

[35]

E. OttJ. SommererJ. AlexanderI. 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.  Google Scholar

[36]

L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.  Google Scholar

[37]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences Cambridge University Press, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar

[38]

N. PlattE. 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.  Google Scholar

[39]

A. N. Shiryayev, Probability Springer Verlag, 1984. doi: 10.1007/978-1-4899-0018-0.  Google Scholar

[40]

J. Stark, Invariant graphs for forced systems, Phys. D, 109 (1997), 163-179.  doi: 10.1016/S0167-2789(97)00167-X.  Google Scholar

[41]

M. Viana, Lectures on Lyapunov Exponents Cambridge University Press, 2014. doi: 10.1017/CBO9781139976602.  Google Scholar

[42]

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.  Google Scholar

[43]

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.  Google Scholar

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 $(K,\infty)$.
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