## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Foundations of Data Science
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
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- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
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- Mathematics in Engineering

### Open Access Journals

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-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

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.

DCDS-B

An experimental conjecture on the existence of
positive periodic solutions for the Brillouin electron beam focusing
system $x''+a(1+\cos2t)x=\frac{1}{x}$
for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.

DCDS-B

This paper is devoted to the numerical approximation of attractors. For general nonautonomous dynamical systems we first introduce a new
type of attractor which includes some classes of noncompact attractors such as
unbounded unstable manifolds. We then adapt two cell mapping algorithms
to the nonautonomous setting and use the computer program GAIO for the
analysis of an explicit example, a two-dimensional system of nonautonomous
difference equations. Finally we present numerical data which indicate a bifurcation of nonautonomous attractors in the Duffing-van der Pol oscillator.

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.

## Year of publication

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