Dichotomy Spectrum Research Articles

The relation between μ-dichotomy spectrum and Sacker-Sell spectrum

We present a relation between μ-dichotomy spectrum and Sacker-Sell spectrum. From this relation, we show that, under some suitable conditions, if the system x′=A(t)x admits a μ-dichotomy, then it admits an exponential dichotomy. Utilizing this result, we present a linearization theorem. Furthermore, we prove that the μ-dichotomy spectrum of the system x′=diag(A1(t),A2(t))x is the union of μ-dichotomy spectra of the subsystems x1′=A1(t)x1 and x2′=A2(t)x2, while this result is wrong if we replace μ-dichotomy spectrum with nonuniform exponential dichotomy spectrum. We also provide a new method to prove the μ-dichotomy spectral theorem.

Nonuniform μ-dichotomy spectrum and kinematic similarity

For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate μ in a large family of growth rates, we consider a notion of spectrum, named nonuniform μ-dichotomy spectrum. This family of spectrums contain the nonuniform dichotomy spectrum as the very particular case of exponential growth rates. For each growth rate μ, we describe all possible forms of the nonuniform μ-dichotomy spectrum, relate its connected components with adapted notions of Lyapunov exponents, and use it to obtain a reducibility result for nonautonomous linear differential equations. We also give illustrative examples where the spectrum is obtained, including a situation where a normal form is obtained for polynomial behavior.

Nonautonomous Fold Bifurcations from Spectral Intervals of Higher Strict Multiplicity

Abstract We provide two sufficient criteria for the bifurcation of bounded entire or homoclinic solutions to nonautonomous difference equations depending on a single real parameter. Our analysis is based on a nonhyperbolic solution, whose variational equation possesses exponential dichotomies on semiaxes ensuring that the corresponding critical spectral interval of the dichotomy spectrum has strict multiplicity > 1. This extends earlier results on the fold bifurcation.

Is the Sacker–Sell type spectrum equal to the contractible set?

For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.

Angular Values of Nonautonomous Linear Dynamical Systems: Part II – Reduction Theory and Algorithm

This work focuses on angular values of nonautonomous dynamical systems which have been introduced for general random and (non)autonomous dynamical systems in a previous publication [W.-J. Beyn, G. Froyland, and T. H\"uls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245--1286]. The angular value of dimension $s$ measures the maximal average rotation which an $s$-dimensional subspace of the phase space experiences through the dynamics of a discrete-time linear system. Our main results relate the notion of angular value to the well-known dichotomy (or Sacker--Sell) spectrum and its associated spectral bundles. In particular, we prove a reduction theorem which shows that instead of maximizing over all subspaces, it suffices to maximize over so-called trace spaces which have their basis in the spectral fibers. The reduction leads to an algorithm for computing angular values of dimensions one and two. We apply the algorithm to several systems of dimension up to 4 and demonstrate its efficiency to detect the fastest rotating subspace even if it is not dominant under the forward dynamics.

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.

Proportional Local Assignability of the Dichotomy Spectrum of One-Sided Discrete Time-Varying Linear Systems

We consider a problem of assignability of the dichotomy spectrum for one-sided discrete time-varying linear systems. Our purpose is to prove that uniform complete controllability is a sufficient condition for proportional local assignability of the dichotomy spectrum.

Sufficient conditions for assignability of nonuniform dichotomy spectrum of discrete time-varying linear systems

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds.

Comments on Poincaré theorem for quasi-periodic systems

<p style='text-indent:20px;'>We establish Poincaré type theorems of normal forms for quasi-periodic systems near the invariant torus based on the exponential dichotomy spectra of linear parts at the normal direction.</p>

Proportional local assignability of dichotomy spectrum of one-sided continuous time-varying linear systems

We consider a local version of the assignment problem for the dichotomy spectrum of linear continuous time-varying systems defined on the half-line. Our aim is to show that uniform complete controllability is a sufficient condition to place the dichotomy spectrum of the closed-loop system in an arbitrary position within some Hausdorff neighborhood of the dichotomy spectrum of the free system using an appropriate time-varying linear feedback. Moreover, we assume that the norm of the matrix of the linear feedback should be bounded from above by the Hausdorff distance between these two spectra with some constant multiplier.

Nonuniform dichotomy spectrum and reducibility for nonautonomous difference equations

Abstract For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

Necessary and Sufficient Conditions for Assignability of Dichotomy Spectrum of One-Sided Discrete Time-Varying Linear Systems

We consider a version of the pole placement problem for one-sided linear discrete time-varying linear systems. Our purpose is to prove that uniform complete controllability is equivalent to possibility of arbitrary assignment of the dichotomy spectrum.

Necessary and sufficient conditions for assignability of dichotomy spectra of continuous time-varying linear systems

We consider a version of the pole placement problem for continuous time-varying linear systems. Our purpose is to prove that uniform complete controllability is equivalent to possibility of arbitrary assignment of the dichotomy spectrum. The main ingredients of the proof are the reduction of system to upper triangular form and the use of the concept of uniform complete stabilization. To illustrate the theoretical result, we consider scalar continuous time-varying control systems. For these systems we provide a simple necessary and sufficient condition for uniform complete controllability and if this condition holds, then we construct an explicit control to assign the dichotomy spectrum.

Reduction principle at work

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

Formal and analytic normal forms for non-autonomous difference systems with uniform dichotomy spectrum

In this paper, we extend formal and analytic normal forms from autonomous difference systems to non-autonomous ones based on the uniform dichotomy spectrum of their linear part.

Nonuniform almost reducibility of nonautonomous linear differential equations

We prove that a linear nonautonomous differential system with nonuniform hyperbolicity on the half line can be written as diagonal system with a perturbation which is small enough. Moreover we show that the diagonal terms are contained in the nonuniform exponential dichotomy spectrum. For this purpose we introduce the concepts of nonuniform almost reducibility and nonuniform contractibility which are generalizations of these notions originally defined in a uniform context.

Assignability of dichotomy spectra for discrete time-varying linear control systems

In this paper, we show that for discrete time-varying linear control systems uniform complete controllability implies arbitrary assignability of dichotomy spectrum of closed-loop systems. This result significantly strengthens the result in [ 5 ] about arbitrary assignability of Lyapunov spectrum of discrete time-varying linear control systems.

A spectral dichotomy version of the nonautonomous Markus–Yamabe conjecture

In this article we introduce a nonautonomous version of the Markus–Yamabe conjecture from an exponential dichotomy spectrum point of view. We prove the validity of this conjecture for the scalar and triangular case. Additionally we show that the origin is a global attractor for an autonomous system by using nonautonomous dynamical systems tools.

A new method to prove the nonuniform dichotomy spectrum theorem in ℝⁿ

This paper presents a new method to prove the nonuniform dichotomy spectrum theorem. Chu et al. [Bull. Sci. Math. 139 (2015), pp. 538–557] and Zhang [J. Funct. Anal. 267 (2014), pp. 1889–1916] generalized the dichotomy spectrum in Siegmund [J. Dynam. Differential Equations 14 (2002), pp. 243–258] to the nonuniform dichotomy spectrum and the authors in these works employed linear integral manifolds (stable and unstable) to establish the spectral theorem. They then used the spectrum theorem to study reducibility. We prove the nonuniform dichotomy spectrum by way of contradiction. In particular, we employ the nonuniform kinematically similarity (nonuniform reducibility) to reduce the shift system into two blocks and then we get a contradiction based on a technique in mathematical analysis. The method in the proof is completely different from previous works.

Conditioned Lyapunov exponents for random dynamical systems

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context. The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the local stability behaviour of trajectories that remain within a bounded domain and - in particular - that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced and its main characteristics are established.