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Author: Siegmund, Stefan × End date: 2019 × Start date: 2000 ×

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Now showing 1 - 5 of 5
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    On Almost Automorphic Dynamics in Symbolic Lattices
    (Georgia Institute of Technology, 2002) Berger, Arno ; Siegmund, Stefan ; Yi, Yingfei
    We study the existence, structure, and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern formation and spatial chaos in infinite dimensional lattice systems are considered, and the construction of chaotic almost automorphic signals is discussed.
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    On the Gap between Random Dynamical Systems and Continuous Skew Products
    (Georgia Institute of Technology, 2002) Berger, Arno ; Siegmund, Stefan
    We review the recent notion of a nonautonomous dynamical system (NDS), which has been introduced as an abstraction of both random dynamical systems and continuous skew product flows. Our focus is on fundamental analogies and discrepancies brought about by these two classes of NDS. We discuss base dynamics mainly through almost periodicity and almost automorphy, and we emphasize the importance of these concepts for NDS which are generated by differential and difference equations. Nonautonomous dynamics is presented by means of representative examples. We also mention several natural yet unresolved questions.
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    Gevrey Regularity of Global Attractor for Generalized Benjamin-Bona-Mahony Equation
    (Georgia Institute of Technology, 2002) Chueshov, I. ; Polat, M. ; Siegmund, Stefan
    We prove the Gevrey regularity of the global attractor of the dynamical system generated by the generalized Benjamin-Bona-Mahony equation with periodic boundary conditions. This result means that elements of the attractor are real analytic functions in spatial variables. As an application we prove the existence of two determining nodes for the problems in one spatial dimension.
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    A Spectral Characterization of Exponential Stability for Linear Time-Invariant Systems on Time Scales
    (Georgia Institute of Technology, 2001) Pötzsche, Christian ; Siegmund, Stefan ; Wirth, Fabian
    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.
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    Pullback Attracting Inertial Manifolds for Nonautonomous Dynamical Systems
    (Georgia Institute of Technology, 2001) Kosch, Norbert ; Siegmund, Stefan
    In this paper we present an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and two additional technical assumptions, called boundedness and coercivity property. Moreover we give conditions which ensure that the global pullback attractor is contained in the inertial manifold. In the second part of the paper we consider special nonautonomous dynamical systems, namely two-parameter semi-flows. As a first application of our abstract approach and for reason of comparison with known results we verify the assumptions for semilinear nonautonomous evolution equations whose linear part satisfies an exponential dichotomy condition and whose nonlinear part is globally bounded and globally Lipschitz.