## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
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### Open Access Journals

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.

$α∈ (0,1)$ |

PROC

Please refer to Full Text.

DCDS

Converse Lyapunov theorems are presented for nonautonomous systems
modelled as skew product flows. These characterize various types
of stability of invariant sets and pullback, forward and uniform
attractors in such nonautonomous systems.

CPAA

In this paper we extend a method to control the dynamics
of evolution equations by finite dimensional controllers which was
suggested by Brunovsky [3] to nonautonomous evolution
equations using nonautonomous inertial manifold theory.

DCDS-B

We prove a general result on the existence of periodic trajectories of systems
of difference equations with finite state space which are phase-locked on
certain components which correspond to cycles in the coupling structure. A main
tool is the new notion of order-induced graph which is similar in spirit to a
Lyapunov function. To develop a coherent theory we introduce the notion of
dynamical systems on finite graphs and show that various existing neural
networks, threshold networks, reaction-diffusion automata and Boolean monomial
dynamical systems can be unified in one parametrized class of dynamical systems
on graphs which we call threshold networks with refraction. For an explicit
threshold network with refraction and for explicit cyclic automata networks
we apply our main result to show the existence
of phase-locked solutions on cycles.

keywords:
discrete dynamical system.
,
directed
graph
,
phase-locking
,
synchronization
,
partial order
,
Neural network

DCDS

We prove a necessary and sufficient condition for the exponential
stability of time-invariant linear systems on time scales in terms of
the eigenvalues of the system matrix. In particular, this unifies the
corresponding characterizations for finite-dimensional differential
and difference equations. To this end we use a representation formula
for the transition matrix of Jordan reducible systems in the
regressive case. Also we give conditions under which the obtained
characterizations can be exactly calculated and explicitly calculate
the region of stability for several examples.

DCDS

We discuss the relationship between invariant manifolds of
nonautonomous differential equations and pullback attractors. This
relationship is essential, e.g., for the numerical approximation
of these manifolds. In the first step, we show that the unstable
manifold is the pullback attractor of the differential equation.
The main result says that every (hyperbolic or nonhyperbolic)
invariant manifold is the pullback attractor of a related system
which we construct explicitly using spectral transformations. To
illustrate our theorem, we present an application to the Lorenz
system and approximate numerically the stable as well as the
strong stable manifold of the origin.

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 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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