Random minimality and continuity of invariant graphs in random dynamical systems
HTML articles powered by AMS MathViewer
- by T. Jäger and G. Keller PDF
- Trans. Amer. Math. Soc. 368 (2016), 6643-6662 Request permission
Abstract:
We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the fibres. For this purpose, we provide suitable random versions of the semiuniform ergodic theorem and also introduce and discuss some basic concepts of random topological dynamics.References
- N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. Rev. E 51 (1995), 980–994.
- K. Pyragas, Weak and strong synchronization of chaos, Phys. Rev. E 54 (1996), R4508–R4511.
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044, DOI 10.1017/CBO9780511755743
- G. Keller, H. Jafri, and R. Ramaswamy, Nature of weak generalized synchronization in chaotically driven maps, Physical Review E 87 (2013), 042913.
- A. J. Homburg, Synchronization in iterated function systems, preprint, 2013.
- Awadhesh Prasad, Surendra Singh Negi, and Ramakrishna Ramaswamy, Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), no. 2, 291–309. MR 1830343, DOI 10.1142/S0218127401002195
- R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity 13 (2000), no. 1, 113–143. MR 1734626, DOI 10.1088/0951-7715/13/1/306
- V. Anagnostopoulou and T. Jäger, Nonautonomous saddle-node bifurcations: random and deterministic forcing, J. Differential Equations 253 (2012), no. 2, 379–399. MR 2921199, DOI 10.1016/j.jde.2012.03.016
- Jaroslav Stark, Regularity of invariant graphs for forced systems, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 155–199. MR 1677161, DOI 10.1017/S0143385799126555
- Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
- V. Anagnostopoulou, T. Jäger, and G. Keller, A model for the non-autonomous hopf bifurcation, Nonlinearity 28 (2015), 2587–2616.
- Sebastian J. Schreiber, On growth rates of subadditive functions for semiflows, J. Differential Equations 148 (1998), no. 2, 334–350. MR 1643183, DOI 10.1006/jdeq.1998.3471
- Yongluo Cao, On growth rates of sub-additive functions for semi-flows: determined and random cases, J. Differential Equations 231 (2006), no. 1, 1–17. MR 2287874, DOI 10.1016/j.jde.2006.08.016
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
- Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
- Yongluo Cao, Stefano Luzzatto, and Isabel Rios, Uniform hyperbolicity for random maps with positive Lyapunov exponents, Proc. Amer. Math. Soc. 136 (2008), no. 10, 3591–3600. MR 2415043, DOI 10.1090/S0002-9939-08-09347-7
- Hans Crauel, Random probability measures on Polish spaces, Stochastics Monographs, vol. 11, Taylor & Francis, London, 2002. MR 1993844
- Gerhard Keller, Equilibrium states in ergodic theory, London Mathematical Society Student Texts, vol. 42, Cambridge University Press, Cambridge, 1998. MR 1618769, DOI 10.1017/CBO9781107359987
- H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601. MR 133429, DOI 10.2307/2372899
Additional Information
- T. Jäger
- Affiliation: Institut für Analysis, Technische Universität Dresden, Zellescher Weg 12-14, 01069 Dresden, Germany
- G. Keller
- Affiliation: Department Mathematik, Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
- Received by editor(s): July 9, 2013
- Received by editor(s) in revised form: September 4, 2014
- Published electronically: December 18, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6643-6662
- MSC (2010): Primary 37A30, 37H15, 34D45
- DOI: https://doi.org/10.1090/tran/6591
- MathSciNet review: 3461046