Carmen Núñez

OLD AND NEW RESULTS ON STRANGE NONCHAOTIC ATTRACTORS

Classical and new results concerning the topological structure of skew-products semiflows, coming from nonautonomous maps and differential equations, are combined in order to establish rigorous conditions giving rise to the occurrence of strange nonchaotic attractors on 𝕋d × ℝ. A special attention is paid to the relation of these sets with the almost automorphic extensions of the base flow. The scope of the results is clarified by applying them to the Harper map, although they are valid in a much wider context.

The almost periodic type difference equations

In this paper, we study the existence of almost periodic type solutions of difference equations by means of exponential dichotomy and trichotomy of linear difference equations.

Minimal sets in monotone and sublinear skew-product semiflows II: Two-dimensional systems of differential equations

Article history: Received 18 June 2009 Revised 2 December 2009 Available online 8 February 2010 MSC: 37B55 37C65 34C12 39B99 35K57

Uniform Weak Disconjugacy and Principal Solutions for Linear Hamiltonian Systems

The paper analyzes the property of (uniform) weak disconjugacy for nonautonomous linear Hamiltonian systems, showing that it is a convenient replacement for the more restrictive property of disconjugacy. In particular, its occurrence ensures the existence of principal solutions. The analysis of the properties of these solutions provides ample information about the dynamics induced by the Hamiltonian system on the Lagrange bundle.

Nonautonomous linear Hamiltonian systems oscillation, spectral theory and control

Nonautonomous linear Hamiltonian systems.- The rotation number and the Lyapunov index for real nonautonomous linear Hamiltonian systems.- The Floquet coeffcient for nonautonomous linear Hamiltonian systems: Atkinson problems.- The Weyl functions.- Weak disconjugacy for linear Hamiltonian systems.- Nonautonomous control theory. Linear regulator problem and the Kalman-Bucy filter.- Nonautonomous control theory. A general version of the Yakubovich Frequency Theorem.- Nonautonomous control theory. Linear-quadratic dissipative control processes.- Index.- References

Nontangential Limit of the Weyl m-Functions for the Ergodic Schrödinger Equation

This paper deals with the spectral and qualitative problems associated with the one-dimensional ergodic Schrödinger equation. Let A2 be the set of those energies for which the real projective flow admits an invariant linear measure with square integrable density function. On this set we calculate the directional derivative of the Floquet coefficient and prove the existence of a nontangential limit of the Weyl m-functions in the L1-topology. In particular, we verify that the known Dieft–Simon inequality for the derivative of the rotation number obtained from Kotanis theory is in fact an equality. In the bounded orbit case we deduce the uniform boundedness of the Weyl m-functions and obtain necessary and sufficient conditions to assure their uniform convergence.

Nonautonomous Linear Hamiltonian Systems

In this chapter, the framework of analysis of the book is described, and the many foundational facts required for this analysis are stated. The first two sections present fundamental notions and properties of topological dynamics and ergodic theory, as well as basic results concerning spaces of matrices, the Grassmannian and Lagrangian manifolds, and matrix-valued functions. In the third section, the different flows induced by a family of linear Hamiltonian systems varying over a compact metric space (which usually arises in a natural way from a nonautonomous system) are described. Special attention is paid to the skew-product flows induced on the Lagrange bundle and on the bundle whose fiber is given by the set of of symmetric matrices. The last section is devoted to a discussion of one of the most fundamental concepts for the forthcoming analysis: that of exponential dichotomy. The dynamical properties implied by its existence are carefully described, some aspects of the Sacker–Sell perturbation theory are explained, and a somewhat nonstandard analysis of the behavior of the Grassmannian flows in the presence of exponential dichotomy is presented.

Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation

We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions

Nonautonomous Control Theory: Linear-Quadratic Dissipative Control Processes

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