Sacker Sell 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.

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.

Smooth linearization under nonuniform hyperbolicity

For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker–Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method.

NONUNIFORM DICHOTOMY SPECTRUM INTERVALS: THEOREM AND COMPUTATION

In this paper, on the condition of nonuniformly bounded growth, we give the relationship between the nonuniform exponential dichotomy spectrum and the other two classical spectrums (the Lyapunov spectrum and Sacker-Sell spectrum), present their stability under small linear perturbations. The main goal of this paper is to discuss the theory for the computation of these spectrums on the condition of nonuniformly bounded growth, and this extends the work of Dieci and Vleck [<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>], which computes the Lyapunov spectrum and Sacker-Sell spectrum under the condition of bounded. Finally, the numerical examples are given to illustrate and verify the theoretical results.

Weak Topologies for Carathéodory Differential Equations: Continuous Dependence, Exponential Dichotomy and Attractors

We introduce new weak topologies and spaces of Carath\'eodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carath\'eodory vector field $f$ provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set or the whole hull of $f$. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given.

A version of the theorem of Johnson, Palmer, and Sell for quasicompact cocycles

The well-known theorem of Johnson, Palmer, and Sell asserts that the endpoints of the Sacker–Sell spectrum of a given cocycle of invertible matrices over a topological dynamical system (M, f) are realized as Lyapunov exponents with respect to some ergodic invariant probability measure for f. In this note we establish the version of this result for quasicompact cocycles of operators acting on an arbitrary Banach space.

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.

A Sternberg Theorem for Nonautonomous Differential Equations

We show that a hyperbolic nonautonomous differential equation can be smoothly linearized if the associated Sacker–Sell spectrum satisfies a non-resonance condition. This result extends the classical Sternberg theorem to nonautonomous differential equations.

Spectral theory for invertible cocycles under nonuniform hyperbolicity

For a linear cocycle with values in the set of invertible \(d\times d\) matrices over an invariant measurable set whose closure cannot be written as a disjoint union of nonempty compact invariant sets, we give a complete characterization of all possible forms of the nonuniform spectrum. The latter is essentially obtained from the Sacker–Sell spectrum replacing uniform hyperbolicity by nonuniform hyperbolicity.

Continuity of the Sacker–Sell spectrum on the half line

ABSTRACTThe Sacker–Sell (also called dichotomy or dynamical) spectrum is an important notion in the stability theory of nonautonomous dynamical systems. For instance, when dealing with variational equations on the (nonnegative) half line, the set Σ+ determines uniform asymptotic stability or instability of a solution and more general, it is crucial to construct invariant manifolds from the stable hierarchy. Compared to the spectrum associated to dichotomies on the entire line, Σ+ has stronger and more flexible perturbation features. In this paper, we study continuity properties of the Sacker–Sell spectrum by means of an operator-theoretical approach. We provide an explicit example that the generally upper-semicontinuous set Σ+ can suddenly collapse under perturbation, establish continuity on the class of equations with discrete spectrum and identify system classes having a continuous spectrum. These results for instance allow to vindicate numerical approximation techniques.

Computing stable hierarchies of fiber bundles

Stable fiber bundles are important structures for understanding nonautonomous dynamics. These sets have a hierarchical structure ranging from stable to strong stable fibers. First, we compute corresponding structures for linear systems and prove an error estimate. The spectral concept of choice is the Sacker-Sell spectrum that is based on exponential dichotomies. Secondly, we tackle the nonlinear case and propose an algorithm for the numerical approximation of stable hierarchies in nonautonomous difference equations. This method generalizes the contour algorithm for computing stable fibers from [ 38 , 39 ]. It is based on Hadamard's graph transform and approximates fibers of the hierarchy by zero-contours of specific operators. We calculate fiber bundles and illustrate errors involved for several examples, including a nonautonomous Lorenz model.

A note on the theorem of Johnson, Palmer and Sell

The well-known theorem of Johnson, Palmer and Sell asserts that the endpoints of the Sacker–Sell spectrum of a given cocycle \(\mathcal {A}\) over a topological dynamical system (M, f) are realized as Lyapunov exponents with respect to some ergodic invariant probability measure for f. The main purpose of this note is to give an alternative proof of this theorem which uses a more recent and independent result of Cao which formulates sufficient conditions for the uniform hyperbolicity of a given cocyle \(\mathcal {A}\) in terms of the nonvanishing of Lyapunov exponents for \(\mathcal {A}\). We also discuss the possibility of obtaining positive results related to the stability of the Sacker–Sell spectra under the perturbations of the cocycle \(\mathcal {A}\).

Continuity and Invariance of the Sacker–Sell Spectrum

The Sacker–Sell (also called dichotomy or dynamical) spectrum \(\varSigma \) 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 \(\varSigma \) and its dynamically relevant subsets. Our criteria allow to avoid nonautonomous bifurcations due to collapsing spectral intervals and justify numerical approximation schemes for \(\varSigma \).

Topological decoupling and linearization of nonautonomous evolution equations

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.

Nonuniform dichotomy spectrum and reducibility for nonautonomous equations

For nonautonomous linear differential equations with nonuniform hyperbolicity, we introduce a definition for nonuniform dichotomy spectrum, which can be seen as a generalization of Sacker–Sell spectrum. We prove a spectral theorem and use the spectral theorem to prove a reducibility result.

Approximating Lyapunov exponents and Sacker–Sell spectrum for retarded functional differential equations

We consider Lyapunov exponents and Sacker---Sell spectrum for linear, nonautonomous retarded functional differential equations posed on an appropriate Hilbert space. A numerical method is proposed to approximate such quantities, based on the reduction to finite dimension of the evolution family associated to the system, to which a classic discrete QR method is then applied. The discretization of the evolution family is accomplished by a combination of collocation and generalized Fourier projection. A rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra. The efficacy of the results is illustrated with some numerical examples.

Perturbation theory for the approximation of stability spectra by QR methods for sequences of linear operators on a Hilbert space

In this paper, we develop a perturbation analysis for stability spectra (Lyapunov exponents and Sacker–Sell spectrum) for products of operators on a Hilbert space (both real and complex) based upon the discrete QR technique. Error bounds are obtained in both the integrally separated and non-integrally separated cases that correspond to distinct and multiple eigenvalues, respectively, for a single linear operator. We illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation (the time varying analogue of gaps between eigenvalues) determines the sensitivity of the stability spectra to perturbation.

On the error in approximating stability spectra for discrete dynamical systems

Abstract In this work, we develop a perturbation theory for the approximation of stability spectra (Lyapunov exponents and Sacker-Sell spectrum) for discrete time dynamical systems. The approach is based upon QR based methods that transform linear time varying discrete time systems to an upper triangular form that allows for extraction of the stability spectra. We focus on nonlinear planar maps and show how shadowing type error results can be combined with perturbation bounds for stability spectra. The utility of our results is illustrated with numerical experiments for the Hénon and standard maps.

An alternative approach to Sacker–Sell spectral theory

In this article, the Morse spectrum is introduced for linear systems of nonautonomous difference equations. In contrast to the well-known Sacker–Sell spectrum, the exponential growth rates are not characterized for the whole system but for a suitable decomposition, given by the finest Morse decomposition. The theory of nonautonomous Morse decompositions is reviewed, and it is shown that the Sacker–Sell spectrum coincides with the Morse spectrum if the system has bounded growth. The observations in this paper can thus be seen as an alternative approach to Sacker–Sell spectral theory.

Computing Sacker–Sell spectra in Discrete Time Dynamical Systems

In this paper we develop boundary value methods for detecting Sacker–Sell spectra in discrete time dynamical systems. The algorithms are advancements of earlier methods for computing projectors of exponential dichotomies. The first method is based on the projector residual $P^2-P$. If this residual is large, then the difference equation has no exponential dichotomy. Further criterions for detecting Sacker–Sell spectral intervals are the norm of end points and midpoints of the solution of a specific boundary value problem. Refined error estimates for the underlying approximation process are given, and the resulting algorithms are applied to an example with known continuous Sacker–Sell spectrum, as well as to the variational equation along orbits of Hénon's map.