Synchronization, Lyapunov Exponents and Stable Manifolds for Random Dynamical Systems
During the past decades, the question of existence and properties of a random attractor of a random dynamical system generated by an S(P)DE has received considerable attention, for example by the work of Gess and Röckner. Recently some authors investigated sufficient conditions which guarantee synchronization, i.e. existence of a random attractor which is a singleton. It is reasonable to conjecture that synchronization and negativity (or non-positivity) of the top Lyapunov exponent of the system should be closely related since both mean that the system is contracting in some sense. Based on classical results by Ruelle, we formulate positive results in this direction. Finally we provide two very simple but striking examples of one-dimensional monotone random dynamical systems for which 0 is a fixed point. In the first example, the Lyapunov exponent is strictly negative but nevertheless all trajectories starting outside of 0 diverge to \(\infty \) or \(-\infty \). In particular, there is no synchronization (not even locally). In the second example (which is just the time reversal of the first), the Lyapunov exponent is strictly positive but nevertheless there is synchronization.
KeywordsSynchronization Lyapunov exponent Random dynamical system
Mathematics Subject Classification (2010)37D10 37D45 37G35 37H15
- 3.Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin-New York (1977)Google Scholar
- 6.Crauel, H.: Random Probability Measures on Polish Spaces. Stochastics Monographs. CRC Press (2003)Google Scholar
- 13.Vorkastner, I.: Noise dependent synchronization of a degenerate SDE. Stoch. Dyn. 18(1), 1850007, 21 (2018)Google Scholar