100 research outputs found
Cm-smoothness of invariant fiber bundles for dynamic equations on measure chains
We present a new self-contained and rigorous proof of the smoothness of invariant fiber bundles for dynamic equations on measure chains or time scales. Here, an invariant fiber bundle is the generalization of an invariant manifold to the nonautonomous case. Our main result generalizes the “Hadamard-Perron theorem” to the time-dependent, infinite-dimensional, noninvertible, and parameter-dependent case, where the linear part is not necessarily hyperbolic with variable growth rates. As a key feature, our proof works without using complicated technical tools
Global bifurcation of homoclinic trajectories of discrete dynamical systems
We prove the existence of an unbounded connected branch of nontrivial
homoclinic trajectories of a family of discrete nonautonomous asymptotically
hyperbolic systems parametrized by a circle under assumptions involving the
topological properties of the asymptotic stable bundles.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1111.140
Topology and Homoclinic Trajectories of Discrete Dynamical Systems
We show that nontrivial homoclinic trajectories of a family of discrete,
nonautonomous, asymptotically hyperbolic systems parametrized by a circle
bifurcate from a stationary solution if the asymptotic stable bundles
Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch
are twisted in different ways.Comment: 19 pages, canceled the appendix (Properties of the index bundle) in
order to avoid any text overlap with arXiv:1005.207
Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems
This work proposes a detectability condition for linear time-varying systems
based on the exponential dichotomy spectrum. The condition guarantees the
existence of an observer, whose gain is determined only by the unstable modes
of the system. This allows for an observer design with low computational
complexity compared to classical estimation approaches. An extension of this
observer design to a class of nonlinear systems is proposed and local
convergence of the corresponding estimation error dynamics is proven. Numerical
results show the efficacy of the proposed observer design technique
Notes on spectrum and exponential decay in nonautonomous evolutionary equations
We first determine the dichotomy (Sacker-Sell) spectrum for certain nonautonomous linear evolutionary equations induced by a class of parabolic PDE systems. Having this information at hand, we underline the applicability of our second result: If the widths of the gaps in the dichotomy spectrum are bounded away from , then one can rule out the existence of super-exponentially decaying (i.e. slow) solutions of semi-linear evolutionary equations
The Index Bundle and Multiparameter Bifurcation for Discrete Dynamical Systems
We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author
On stability of delay difference equations with variable coefficients: successive products tests
Computation of integral manifolds for Carathéodory differential equations
We derive two numerical approximation schemes for local invariant manifolds of nonautonomous ordinary differential equations which can be measurable in time and Lipschitzian in the spatial variable. Our approach is inspired by previous work of Jolly, Rosa (2005), "Computation of non-smooth local center manifolds", IMA Journal of Numerical Analysis 25, 698-725, on autonomous ODEs and based on truncated Lyapunov-Perron operators. Both of our methods are applicable to the full hierarchy of strongly stable, stable, center-stable and the corresponding unstable manifolds, and exponential refinement strategies yield exponential convergence. Several examples illustrate our approach
- …