Dichotomy spectra of triangular equations
Christian Pötzsche
Discrete & Continuous Dynamical Systems - A 2016, 36(1): 423-450 doi: 10.3934/dcds.2016.36.423
Without question, the dichotomy spectrum is a central tool in the stability, qualitative and geometric theory of nonautonomous dynamical systems. In this context, when dealing with time-variant linear equations having triangular coefficient matrices, their dichotomy spectrum associated to the whole time axis is not fully determined by the diagonal entries. This is surprising because such a behavior differs from both the half line situation, as well as the classical autonomous and periodic cases. At the same time triangular problems occur in various applications and particularly numerical techniques.
    Based on operator-theoretical tools, this paper provides various sufficient and verifiable 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.
keywords: exponential dichotomy nonautonomous hyperbolicity Bohl exponent block triangular equation Sacker-Sell spectrum spectral continuity. diagonal significance Dichotomy spectrum
Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern
Christian Pötzsche
Discrete & Continuous Dynamical Systems - A 2011, 31(3): 941-973 doi: 10.3934/dcds.2011.31.941
This paper continues our work on local bifurcations for nonautonomous difference and ordinary differential equations. Here, it is our premise that constant or periodic solutions are replaced by bounded entire solutions as bifurcating objects in order to encounter right-hand sides with an arbitrary time dependence.
    We introduce a bifurcation pattern caused by a dominant spectral interval (of the dichotomy spectrum) crossing the stability boundary. As a result, differing from the classical autonomous (or periodic) situation, the change of stability appears in two steps from uniformly asymptotically stable to asymptotically stable and finally to unstable. During the asymptotically stable regime, a whole family of bounded entire solutions occurs (a so-called "shovel"). Our basic tools are exponential trichotomies and a quantitative version of the surjective implicit function theorem yielding the existence of strongly center manifolds.
keywords: strongly center fiber bundle shovel bifurcation dichotomy spectrum strongly center integral manifold. nonautonomous difference equation nonautonomous differential equation exponential trichotomy Nonautonomous bifurcation
Kenneth J. Palmer Christian Pötzsche
Discrete & Continuous Dynamical Systems - B 2017, 22(8): i-ii doi: 10.3934/dcdsb.201708i
Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach
Christian Pötzsche
Discrete & Continuous Dynamical Systems - B 2010, 14(2): 739-776 doi: 10.3934/dcdsb.2010.14.739
We investigate local bifurcation properties for nonautonomous difference and ordinary differential equations. Extending a well-established autonomous theory, due to our arbitrary time dependence, equilibria or periodic solutions typically do not exist and are replaced by bounded complete solutions as possible bifurcating objects.
   Under this premise, appropriate exponential dichotomies in the variational equation along a nonhyperbolic solution on both time axes provide the necessary Fredholm theory in order to employ a Lyapunov-Schmidt reduction. Among other results, this yields nonautonomous versions of the classical fold, transcritical and pitchfork bifurcation patterns.
keywords: Fredholm operator nonautonomous bifurcation complete solution dichotomy spectrum. nonautonomous difference equation nonautonomous differential equation Lyapunov-Schmidt reduction
Smooth roughness of exponential dichotomies, revisited
Christian Pötzsche
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 853-859 doi: 10.3934/dcdsb.2015.20.853
As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.
keywords: roughness Exponential dichotomy smooth parameter-dependence nonautonomous dynamical system.
Topological decoupling and linearization of nonautonomous evolution equations
Christian Pötzsche Evamaria Russ
Discrete & Continuous Dynamical Systems - S 2016, 9(4): 1235-1268 doi: 10.3934/dcdss.2016050
Topological linearization results typically require solution flows rather than merely semiflows. An exception occurs when the linearization fulfills spectral assumptions met e.g. for scalar reaction-diffusion equations. We employ tools from the geometric theory of nonautonomous dynamical systems in order to extend earlier work by Lu [12] to time-variant evolution equations under corresponding conditions on the Sacker-Sell spectrum of the linear part. Our abstract results are applied to nonautonomous reaction-diffusion and convection equations.
keywords: dichotomy spectrum. integral manifolds Topological linearization reaction-diffusion equation invariant foliations topological decoupling asymptotic phase
Nonautonomous continuation of bounded solutions
Christian Pötzsche
Communications on Pure & Applied Analysis 2011, 10(3): 937-961 doi: 10.3934/cpaa.2011.10.937
We show the persistence of hyperbolic bounded solutions to nonautonomous difference and retarded functional differential equations under parameter perturbation, where hyperbolicity is given in terms of an exponential dichotomy in variation. Our functional-analytical approach is based on a formulation of dynamical systems as operator equations in ambient sequence or function spaces. This yields short proofs, in particular of the stable manifold theorem.
As an ad hoc application, the behavior of hyperbolic solutions and stable manifolds for ODEs under numerical discretization with varying step-sizes is studied.
keywords: hyperbolicity nonautonomous continuation stable manifold functional differential equation Difference equation dichotomy spectrum exponential dichotomy surjective implicit function theorem discretization.
A spectral characterization of exponential stability for linear time-invariant systems on time scales
Christian Pötzsche Stefan Siegmund Fabian Wirth
Discrete & Continuous Dynamical Systems - A 2003, 9(5): 1223-1241 doi: 10.3934/dcds.2003.9.1223
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.

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