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On the normalized ground states for the Kawahara equation and a fourth order NLS
Synchronisation of almost all trajectories of a random dynamical system
Department of Mathematics, University of Exeter, Exeter, EX4 4QF, United Kingdom |
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).
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. Google Scholar |
[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. Google Scholar |
[18] |
J. Newman, Synchronisation in invertible random dynamical systems on the circle, preprint, arXiv: 1502.07618v2. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[18] |
J. Newman, Synchronisation in invertible random dynamical systems on the circle, preprint, arXiv: 1502.07618v2. Google Scholar |
[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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