DCDS-B
On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems
Doan Thai Son
Discrete & Continuous Dynamical Systems - B 2017, 22(8): 3113-3126 doi: 10.3934/dcdsb.2017166

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
DCDS-B
An instability theorem for nonlinear fractional differential systems
Nguyen Dinh Cong Doan Thai Son Stefan Siegmund Hoang The Tuan
Discrete & Continuous Dynamical Systems - B 2017, 22(8): 3079-3090 doi: 10.3934/dcdsb.2017164
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
$\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$
where
$α∈ (0,1)$
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
DCDS-B
Nonautonomous finite-time dynamics
Arno Berger Doan Thai Son Stefan Siegmund
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 463-492 doi: 10.3934/dcdsb.2008.9.463
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

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