Stefan Siegmund

Dichotomy Spectrum for Nonautonomous Differential Equations

For nonautonomous linear differential equations ⋅x=A(t) x with locally integrable A: R→RN×N the so-called dichotomy spectrum is investigated in this paper. As the closely related dichotomy spectrum for skew product flows with compact base (Sacker–Sell spectrum) our dichotomy spectrum for nonautonomous differential equations consists of at most N closed intervals, which in contrast to the Sacker–Sell spectrum may be unbounded. In the constant coefficients case these intervals reduce to the real parts of the eigenvalues of A. In any case the spectral intervals are associated with spectral manifolds comprising solutions with a common exponential growth rate. The main result of this paper is a spectral theorem which describes all possible forms of the dichotomy spectrum.

The dichotomy spectrum for noninvertible systems of linear difference equations

In this paper we introduce the so—called dichotomy spectrum for nonautonomous linear difference equations , whose coefficient matrices are not supposed to be invertible. This new kind of spectrum is based on the notion of exponential forward dichotomy and consists of at most N+ 1 closed, not necessarily bounded, intervals of the positive real line. If all the matrices A(k) are invertible then the number of spectral intervals is at most N, and if in addition A(k) ≡ A is independent of k then these intervals reduce to the absolut values of the eigenvalues of A. In any case the spectral intervals are associated with invariant vector bundles comprising solutions with a common exponential growth rate. The main result of this paper is a Spectral Theorem which describes all possible forms of the dichotomy spectrumIn this paper we introduce the so—called dichotomy spectrum for nonautonomous linear difference equations , whose coefficient matrices are not supposed to be invertible. This new kind of spectrum i...

BIFURCATIONS AND CONTINUOUS TRANSITIONS OF ATTRACTORS IN AUTONOMOUS AND NONAUTONOMOUS SYSTEMS

Nonautonomous bifurcation theory studies the change of attractors of nonautonomous systems which are introduced here with the process formalism as well as the skew product formalism. We present a t...

A new class of three-point methods with optimal convergence order eight and its dynamics

We establish a new class of three-point methods for the computation of simple zeros of a scalar function. Based on the two-point optimal method by Ostrowski (1966), we construct a family of order eight methods which use three evaluations of f and one of f′ and therefore have an efficiency index equal to 84≈1.682

HYPERBOLICITY AND INVARIANT MANIFOLDS FOR PLANAR NONAUTONOMOUS SYSTEMS ON FINITE TIME INTERVALS

\sqrt [4]{8}\approx 1.682

On stable manifolds for planar fractional differential equations

and are optimal in the sense of the Kung and Traub conjecture (Kung and Traub J. Assoc. Comput. Math. 21, 634–651, 1974). Moreover, the dynamics of the proposed methods are shown with some comparisons to other existing methods. Numerical comparison with existing optimal schemes suggests that the new class provides a valuable alternative for solving nonlinear equations.

On stability of linear time-varying second-order differential equations

The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from fixed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lyapunov–Perron approach or Hadamards graph transformation. We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are defined or known only for a finite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the finite time interval. We prove an analog of the Theorem of Linearized Asymptotic Stability on finite time intervals, generalize the Okubo–Weiss criterion from fluid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre flow and symmetric vortex merger.

Realization of Try-Once-Discard in Wireless Multihop Networks

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov-Perron operator. An example is provided to illustrate the result.

General analysis of mathematical models for bone remodeling

We derive sufficient conditions for stability and asymptotic stability of second order, scalar differential equations with differentiable coefficients.

Pullback Attracting Inertial Manifolds for Nonautonomous Dynamical Systems

In networked control systems, the Try-Once-Discard (TOD) protocol is of high interest because its properties can be characterized by Lyapunov functions. This feature makes it practical to incorporate TOD into Lyapunov-based design of linear and nonlinear control systems, yielding a self-contained theory for system stabilization. In previous work, candidates for TOD realizations for single-hop (wired and wireless) networks have been proposed. However, it has been a hitherto open question whether TOD can be realized in wireless multihop networks. In fact, it is far from obvious how dynamic value-based competition with deterministically bounded maximum delay, as required by TOD, is achievable in wireless multihop networks. In this paper, we give a positive answer to this question, by presenting a functionally complete realization of TOD in wireless multihop networks. Our solution is based on highly accurate multihop tick synchronization, and applies an algorithm for collision-protected network-wide value arbitration with deterministic delay. We provide experimental evidence for the feasibility of our solution on existing micro controller platforms, and assess our TOD realization in a batch reactor scenario.