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 and 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.

[2]

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

[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. 

[4]

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

[5]

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

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[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. 

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[32]

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

[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.

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[40]

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

[41]

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

[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.

[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.

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.

[2]

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

[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. 

[4]

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

[5]

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

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[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. 

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

[30]

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

[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.

[32]

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

[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.

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[40]

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

[41]

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

[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.

[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.

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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