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
Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations
Stefan Siegmund
Please refer to Full Text.
keywords: nonautonomous normal form parametric perturbation. Duffing-van der Pol oscillator Generalized Poincare normal form
Lyapunov's second method for nonautonomous differential equations
Lars Grüne Peter E. Kloeden Stefan Siegmund Fabian R. Wirth
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.
keywords: nonautonomous attractor. nonautonomous differential equation Lyapunov's second method nonautonomous dynamical system Lyapunov function stability
Feedback control via inertial manifolds for nonautonomous evolution equations
Norbert Koksch Stefan Siegmund
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.
keywords: feed-back control process. Inertial manifold nonautonomous dynamical systems
Phase-locked trajectories for dynamical systems on graphs
Jeremias Epperlein Stefan Siegmund
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
A spectral characterization of exponential stability for linear time-invariant systems on time scales
Christian Pötzsche Stefan Siegmund Fabian Wirth
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.
keywords: Time scale linear dynamic equation exponential stability.
Invariant manifolds as pullback attractors of nonautonomous differential equations
Bernd Aulbach Martin Rasmussen Stefan Siegmund
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.
keywords: invariant manifold pullback attractor numerical approximation. Nonautonomous differential equation
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
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.
Positive periodic solution for Brillouin electron beam focusing system
Jingli Ren Zhibo Cheng Stefan Siegmund
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.
keywords: Brillouin electron beam focusing system singularity. Liénard equation

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