Dichotomies and spectra
AAU
A note on the dichotomy spectrum
Journal of Difference Equations and Applications 15(10), 2009,
1021-1025
In many ways, exponential dichotomies are an appropriate hyperbolicity notion for nonautonomous linear differential or difference equations. The corresponding dichotomy spectrum generalizes the classical set of eigenvalues or Floquet multipliers and is therefore of eminent importance in a stability theory for explicitly time-dependent systems, as well as to establish a geometric theory of nonautonomous problems with ingredients like invariant manifolds and normal forms, or to deduce continuation and bifurcation techniques.
In this note, we derive some invariance and perturbation properties of the dichotomy spectrum for nonautonomous linear difference equations in Banach spaces. They easily follow from the observation that the dichotomy spectrum is strongly related to a weighted shift operator on an ambient sequence space.
A corrigendum appeared in Journal of Difference Equations and Applications 18(7), 2012, 1257-1261
Christian Pötzsche (Jan 2018)
Fine structure of the dichotomy spectrum
Integral Equations and Operator Theory 73(1), 2012, 107-151
The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it yields information on stability, robustness and bifurcation properties. This paper explores a helpful connection between the dichotomy spectrum and operator theory. Indeed, it relates the asymptotic behavior of a linear nonautonomous difference equation to the spectrum of weighted shift operators. This link enables us to define and study several subsets of the dichotomy spectrum providing a more detailed insight with promising applications to a bifurcation theory for difference equations with explicitly time-dependent right-hand sides.
Smooth roughness of exponential dichotomies,
Discrete and Continuous Dynamical Systems, Series B, 20(3), 2015, 853-859
In their recent paper [Proc. Am. Math. Soc. 139 (2011), no. 3, 999-1012], Barreira and Valls show that the invariant projectors of exponential dichotomies, and therefore the associated stable and unstable vector bundles, depend continuously on parameters, provided the perturbation is small in the $C^1$-topology.
We give a direct and independent proof of this result and moreover enhance it in various aspects
Dichotomy spectra of triangular equations
Discrete and Continuous Dynamical Systems, Series A, 36(1), 2016, 423-450
Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. However, when dealing with such linear equations having triangular coefficient matrices, their dichotomy spectrum associated to the whole time axis is not fully determined by the diagonal entries. On the one hand, this might be surprising because such behavior differs from both the half line situation, as well as the classical autonomous and periodic cases. On the other hand, triangular problems surely occur in various applications and numerical techniques.
Based on operator-theoretical tools, this paper provides various sufficient criteria to obtain a corresponding diagonal significance for finite-dimensional difference equations in the following sense: Spectral and continuity properties of the diagonal elements extend to the whole triangular system.
Continuity and invariance of the Sacker-Sell
(with E. Russ) Journal of Dynamics and Differential Equations 28(2), 2016, 533-566
The dichotomy (also called Sacker-Sell or dynamical) spectrum $\Sigma$ is a fundamental concept in the geometric, as well as for a developing bifurcation theory of nonautonomous dynamical systems. In general, it behaves merely upper-semicontinuously and a perturbation theory is therefore delicate. This paper explores an operator-theoretical approach to obtain invariance and continuity conditions for both $\Sigma$ and its dynamically relevant subsets. Our criteria allow to avoid nonautonomous bifurcations due to collapsing spectral intervals and justify numerical approximation schemes for $\Sigma$.
Notes on spectrum and exponential decay in
nonautonomous evolutionary equations
(with E. Russ) Electronic Journal of Qualitative Theory of Differerential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ. (July 1-4, 2015, Szeged, Hungary)
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 $0$, then one can rule out the existence of super-exponentially decaying (i.e. slow) solutions of semi-linear evolutionary equations.
Continuity of the Sacker-Sell spectrum on the
Dynamical Systems 33(1), 2018, 27-53
The dichotomy (also called Sacker-Sell or dynamical) spectrum $\Sigma^+\subseteq\R$ is a central notion in the stability theory of nonautonomous dynamical systems. For instance, when dealing with variational equations on the (nonnegative) half line, the set $\Sigma^+$ determines uniform asymptotic stability or instability of a solution and more general, it turned out to be crucial to construct invariant manifolds from the stable hierarchy. In this paper, we study continuity properties of the dichotomy spectrum by means of an operator-theoretical approach. Compared to the spectrum associated to dichotomies on the whole time axis, $\Sigma^+$ apparently has stronger and more flexible perturbation properties. Given the generally upper-semicontinuous set $\Sigma^+$, our results for instance allow to vindicate numerical approximation techniques.
Dichotomy spectra of nonautonomous linear
in “Advances in Difference Equations and Discrete Dynamical Systems”, S. Elaydi, Y. Hamaya, H. Matsunaga, C. Pötzsche eds., Springer Proceedings in Mathematics & Statistics 212, 27--54, 2017
We give examples of dichotomy spectra for nonautonomous linear difference equations in infinite-dimensional spaces. Particular focus is on the spectrum of integrodifference equations having compact coefficients. Concrete systems with explicitly known spectra are discussed for several reasons: (1) They yields reference examples for numerical approximation schemes. (2) The asymptotic behavior of spectral intervals is tackled illustrating their merging.
