Explore the vast range of titles available.

The topological structure of ‘mean dichotomy spectrum’ is shown in this article, as an extension of Sacker–Sell spectrum and non-uniform dichotomy spectrum. With regard to mean hyperbolic systems, the coexistence of expansion and contraction behaviours can lead to non-hyperbolic phenomena during evolution process. To be precise, distinct from uniform and non-uniform hyperbolic cases, error hyperbolic degree$\\varepsilon(t,\\tau)$is vital to depict the spectral manifolds. As application, the reducibility theorem for mean hyperbolic systems is provided to deduce block diagonalization.

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.

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.

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.

A novel method for separating the matrix spectrum by a straight line based on a fractional linear transformation is proposed. This method has a number of advantages over the approaches based on an exponential transformation; more precisely, the range of its application is wider and the number of iterations needed for its convergence is much lower. The proposed method is used to study flutter problems for an infinite strip under various edge fastening conditions, which, after suitable discretization of differential operators, are reduced to spectral problems for linear operators. The study of stability regions by the method of spectrum dichotomy by the imaginary axis makes it possible to construct neutral curves in the plane of parameters of the flutter problem.

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.

For a linear flow Φ\\Phi on a vector bundle π:E→S\\pi : E \\rightarrow S a spectrum can be defined in the following way: For a chain recurrent component M\\mathcal {M} on the projective bundle PE\\mathbb {P} E consider the exponential growth rates associated with (finite time) (ε,T)(\\varepsilon ,T)-chains in M\\mathcal {M}, and define the Morse spectrum ΣMo(M,Φ)\\Sigma _{Mo}(\\mathcal {M},\\Phi ) over M\\mathcal {M} as the limits of these growth rates as ε→0\\varepsilon \\rightarrow 0 and T→∞T \\rightarrow \\infty. The Morse spectrum ΣMo(Φ)\\Sigma _{Mo}(\\Phi ) of Φ\\Phi is then the union over all components M⊂PE\\mathcal {M}\\subset \\mathbb {P}E. This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec̆ spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that ΣMo(Φ)\\Sigma _{Mo}(\\Phi ) contains all Lyapunov exponents of Φ\\Phi for arbitrary initial values, and the ΣMo(M,Φ)\\Sigma _{Mo}(\\mathcal {M},\\Phi ) are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that Φ\\Phi is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of EE may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in PE\\mathbb {P} E) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledeč, the Lyapunov, and the Morse spectrum. We present an example (flows induced by C∞C^{\\infty } vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen’s Shadowing Lemma.

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

Everyday aesthetics, at its core, is based on the supposed dichotomy between art and life, considering life as something routine-like, and art as the breaking of the routine, something charismatic. Different authors of everyday aesthetics use different words to describe this dichotomy. For example, in his article “What is ‘Everyday’ in Everyday Aesthetics?”, Ossi Naukkarinen simply uses everydayness and non-everyday-like, while Arto Haapala, in his “On the Aesthetics of the Everyday: Familiarity, Strangeness, and the Meaning of Place” uses the terms familiarity and strangeness. The authors also propose different ways of bridging this dichotomy. However, as the paper shows, the real question is not how to bridge the dichotomy itself but rather whether the dichotomy exists in the first place. Moreover, the paper suggests a change of direction in future investigations of everyday aesthetics, and focusing on the nuances that exist on the routine-charisma and charismatic-routine spectrum, supported by academic research and the personal account of the paper’s author art project. Moreover, the implications of this shift extend beyond the boundaries of everyday aesthetics.

In the present article, we consider the problem on location of the matrix spectrum with respect to a parabola. In terms of solvability of a matrix Lyapunov type equation, we prove theorems on location of the matrix spectrum in certain domains (bounded by a parabola) and (lying outside the closure of ). A solution to the matrix equation is constructed. We use this equation and prove an analog of the Lyapunov–Krein theorem on dichotomy of the matrix spectrum with respect to a parabola.