An instability theorem for nonlinear fractional differential systems
Nguyen Dinh Cong Doan Thai Son Stefan Siegmund Hoang The Tuan
In this paper, we give a criterion on instability of an equilibrium of a nonlinear Caputo fractional differential system. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector
$\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$
$α∈ (0,1)$
is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.
keywords: Fractional differential equations qualitative theory stability theory instability condition
On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems
Doan Thai Son

In this paper, we construct an open and dense set in the space of bounded linear random dynamical systems (both discrete and continuous time) equipped with the essential sup norm such that the Lyapunov exponents depend analytically on the coefficients in this set. As a consequence, analyticity for Lyapunov exponents of bounded linear random dynamical systems is a generic property.

keywords: Lyapunov exponents random dynamical systems
Nonautonomous finite-time dynamics
Arno Berger Doan Thai Son Stefan Siegmund
Nonautonomous differential equations on finite-time intervals play an increasingly important role in applications that incorporate time-varying vector fields, e.g. observed or forecasted velocity fields in meteorology or oceanography which are known only for times $t$ from a compact interval. While classical dynamical systems methods often study the behaviour of solutions as $t \to \pm\infty$, the dynamic partition (originally called the EPH partition) aims at describing and classifying the finite-time behaviour. We discuss fundamental properties of the dynamic partition and show that it locally approximates the nonlinear behaviour. We also provide an algorithm for practical computations with dynamic partitions and apply it to a nonlinear 3-dimensional example.
keywords: nonautonomous differential equations on finite-time intervals dynamic partition. Hyperbolicity
The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor
Thai Son Doan Martin Rasmussen Peter E. Kloeden
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
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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