The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor
Thai Son Doan Martin Rasmussen Peter E. Kloeden
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 875-887 doi: 10.3934/dcdsb.2015.20.875
The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.
keywords: mean-square random dynamical system bifurcation. mean-square random attractor Dichotomy spectrum
On Lyapunov exponents of difference equations with random delay
Nguyen Dinh Cong Thai Son Doan Stefan Siegmund
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 861-874 doi: 10.3934/dcdsb.2015.20.861
The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.
keywords: Random difference equations random delay multiplicative ergodic theorem Lyapunov exponent.

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