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July  2020, 40(7): 4163-4177. doi: 10.3934/dcds.2020176

Synchronisation of almost all trajectories of a random dynamical system

Julian Newman

Department of Mathematics, University of Exeter, Exeter, EX4 4QF, United Kingdom

Received  April 2019 Revised  December 2019 Published  April 2020

Fund Project: The author wishes gratefully to acknowledge funding from an EPSRC Doctoral Training Account and an EPSRC Doctoral Prize, both at Imperial College London

It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally asymptotically stable trajectories, then there is a random attracting set consisting of finitely many points, whose basin of forward-time attraction includes a random full measure open set. In this paper, we present necessary and sufficient conditions for this attracting set to be a singleton. Our result does not require the state space to be compact, but holds on general Lusin metric spaces (in both discrete and continuous time).

Keywords: Random dynamical systems, synchronisation, asymptotic stability, invariant measures, Markov processes.
Mathematics Subject Classification: Primary: 37H05; Secondary: 60F15.
Citation: Julian Newman. Synchronisation of almost all trajectories of a random dynamical system. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4163-4177. doi: 10.3934/dcds.2020176
References:
[1]

V. A. Antonov, Modeling of processes of cyclic evolution type. Synchronization by a random signal, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1984, 67–76.

[2]

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

[3]

P. H. Baxendale, Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms, in Spatial Stochastic Processes, Progr. Probab., 19, Birkhäuser Boston, Boston, MA, 1991,189–218. doi: 10.1007/978-1-4612-0451-0_9.

[4]

P. Berti, L. Pratelli and P. Rigo, Almost sure weak convergence of random probability measures, Stochastics, 78 (2006), 91-97.  doi: 10.1080/17442500600745359.

[5]

M. Cranston, B. Gess and M. Scheutzow, Weak synchronization for isotropic flows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3003-3014.  doi: 10.3934/dcdsb.2016084.

[6]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11, Taylor & Francis, London, 2002.

[7]

H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dynam. Differential Equations, 10 (1998), 259-274.  doi: 10.1023/A:1022665916629.

[8]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Related Fields, 168 (2017), 511-556.  doi: 10.1007/s00440-016-0716-2.

[9]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems, Ann. Probab., 45 (2017), 1325-1350.  doi: 10.1214/16-AOP1088.

[10]

D. H. Fremlin, Measure Theory. 4: Topological Measure Spaces, Torres Fremlin, Colchester, 2006.

[11]

A. J. Homburg, Synchronization in iterated function systems, preprint, arXiv: 1303.6054v1.

[12]

T. Kaijser, On stochastic perturbations of iterations of circle maps, Phys. D, 68 (1993), 201-231.  doi: 10.1016/0167-2789(93)90081-B.

[13]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.

[14] S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge Tracts in Mathematics, 194, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139137119.
[15]

Y. Le Jan, \'{E}quilibre statistique pour les produits de difféomorphismes aléatoires indépendants, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 111-120. 

[16]

D. Malicet, Random walks on Homeo(S1), Commun. Math. Phys., 356 (2017), 1083-1116.  doi: 10.1007/s00220-017-2996-5.

[17]

J. Newman, Ergodic theory for semigroups of Markov kernels, 2015. Available from: http://wwwf.imperial.ac.uk/ jmn07/Ergodic_Theory_for_Semigroups_of_Markov_Kernels.pdf.

[18]

J. Newman, Synchronisation in invertible random dynamical systems on the circle, preprint, arXiv: 1502.07618v2.

[19]

J. Newman, Necessary and sufficient conditions for stable synchronization in random dynamical systems, Ergodic Theory Dynam. Systems, 38 (2018), 1857-1875.  doi: 10.1017/etds.2016.109.

[20]

A. S. Pikovskii, Synchronization and stochastization of array of self-excited oscillators by external noise, Radiophys. Quantum Electron., 27 (1984), 390-395.  doi: 10.1007/BF01044784.

[21]

S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, 180, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-85473-6.

[22]

R. Toral, C. R. Mirasso, E. Hernández-García and O. Piro, Analytical and numerical studies of noise-induced synchronization of chaotic systems, Chaos, 11 (2001), 665-673.  doi: 10.1063/1.1386397.

show all references

References:
[1]

V. A. Antonov, Modeling of processes of cyclic evolution type. Synchronization by a random signal, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1984, 67–76.

[2]

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

[3]

P. H. Baxendale, Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms, in Spatial Stochastic Processes, Progr. Probab., 19, Birkhäuser Boston, Boston, MA, 1991,189–218. doi: 10.1007/978-1-4612-0451-0_9.

[4]

P. Berti, L. Pratelli and P. Rigo, Almost sure weak convergence of random probability measures, Stochastics, 78 (2006), 91-97.  doi: 10.1080/17442500600745359.

[5]

M. Cranston, B. Gess and M. Scheutzow, Weak synchronization for isotropic flows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3003-3014.  doi: 10.3934/dcdsb.2016084.

[6]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11, Taylor & Francis, London, 2002.

[7]

H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dynam. Differential Equations, 10 (1998), 259-274.  doi: 10.1023/A:1022665916629.

[8]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Related Fields, 168 (2017), 511-556.  doi: 10.1007/s00440-016-0716-2.

[9]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems, Ann. Probab., 45 (2017), 1325-1350.  doi: 10.1214/16-AOP1088.

[10]

D. H. Fremlin, Measure Theory. 4: Topological Measure Spaces, Torres Fremlin, Colchester, 2006.

[11]

A. J. Homburg, Synchronization in iterated function systems, preprint, arXiv: 1303.6054v1.

[12]

T. Kaijser, On stochastic perturbations of iterations of circle maps, Phys. D, 68 (1993), 201-231.  doi: 10.1016/0167-2789(93)90081-B.

[13]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.

[14] S. Kuksin and A. Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge Tracts in Mathematics, 194, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139137119.
[15]

Y. Le Jan, \'{E}quilibre statistique pour les produits de difféomorphismes aléatoires indépendants, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 111-120. 

[16]

D. Malicet, Random walks on Homeo(S1), Commun. Math. Phys., 356 (2017), 1083-1116.  doi: 10.1007/s00220-017-2996-5.

[17]

J. Newman, Ergodic theory for semigroups of Markov kernels, 2015. Available from: http://wwwf.imperial.ac.uk/ jmn07/Ergodic_Theory_for_Semigroups_of_Markov_Kernels.pdf.

[18]

J. Newman, Synchronisation in invertible random dynamical systems on the circle, preprint, arXiv: 1502.07618v2.

[19]

J. Newman, Necessary and sufficient conditions for stable synchronization in random dynamical systems, Ergodic Theory Dynam. Systems, 38 (2018), 1857-1875.  doi: 10.1017/etds.2016.109.

[20]

A. S. Pikovskii, Synchronization and stochastization of array of self-excited oscillators by external noise, Radiophys. Quantum Electron., 27 (1984), 390-395.  doi: 10.1007/BF01044784.

[21]

S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, 180, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-85473-6.

[22]

R. Toral, C. R. Mirasso, E. Hernández-García and O. Piro, Analytical and numerical studies of noise-induced synchronization of chaotic systems, Chaos, 11 (2001), 665-673.  doi: 10.1063/1.1386397.

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