Lyapunov theorems for Banach spaces

We present a spectral mapping theorem for semigroups on any Banach space $E$. From this, we obtain a characterization of exponential dichotomy for nonautonomous differential equations for $E$-valued functions. This characterization is given in terms of the spectrum of the generator of the semigroup of evolutionary operators.Comment: 6 page

Exponential dichotomies of evolution operators in Banach spaces

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid

On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in 'Nonlinear Dynamics'. The final authenticated version is available online at https://doi.org/10.1007/s11071-020-05620-8

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

In this paper we obtain a detailed description of the global and cocycle attractors for the skew-product semiflows induced by the mild solutions of a family of scalar linear-dissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, basically two different types of attractors can appear, depending on whether the linear coefficient in the equations determines a bounded or an unbounded associated real cocycle. In the first case (the one for periodic equations), the structure of the attractor is simple, whereas in the second case (which happens in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations in which the attractor is chaotic in measure in the sense of Li-Yorke. Besides, we obtain a non-autonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a linear-dissipative version of the Chafee-Infante equation.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalEuropean CommissionJunta de Andalucí