## Journals

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DCDS-B

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

is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.

$α∈ (0,1)$ |

DCDS-B

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.

DCDS-B

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.

DCDS-B

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.

DCDS-B

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.

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