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Spectral theta series of operators with periodic bicharacteristic flow

Jens Marklof (2007)

Annales de l’institut Fourier

The theta series ϑ ( z ) = ∑ exp ( 2 π i n 2 z ) is a classical example of a modular form. In this article we argue that the trace ϑ P ( z ) = Tr exp ( 2 π i P 2 z ) , where P is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of ϑ P ( z ) near the real axis, and the proof of logarithm laws and limit theorems for its value...

Spreadability, Vulnerability and Protector Control

A. Bernoussi (2010)

Mathematical Modelling of Natural Phenomena

In this work, we present some concepts recently introduced in the analysis and control of distributed parameter systems: Spreadability, vulnerability and protector control. These concepts permit to describe many biogeographical phenomena, as those of pollution, desertification or epidemics, which are characterized by a spatio-temporal evolution

SRB-like Measures for C⁰ Dynamics

Eleonora Catsigeras, Heber Enrich (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

For any continuous map f: M → M on a compact manifold M, we define SRB-like (or observable) probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f always has observable measures, even if SRB measures do not exist. We prove that the definition of observability is optimal, provided that the purpose of the researcher is to describe the asymptotic statistics for Lebesgue almost all initial states. Precisely, the never empty set of all observable measures is...

Stabilité des systèmes à commutations du plan

Ugo Boscain, Grégoire Charlot, Mario Sigalotti (2009/2010)

Séminaire de théorie spectrale et géométrie

Soient X et Y deux champs de vecteurs lisses sur ℝ 2 globalement asymptotiquement stables à l’origine. Nous donnons des conditions nécessaires et des conditions suffisantes sur la topologie de l’ensemble des points où X et Y sont parallèles pour pouvoir assurer la stabilité asymptotique globale du système contrôlé non linéaire non autonome q ˙ ( t ) = u ( t ) X ( q ( t ) ) + ( 1 - u ( t ) ) Y ( q ( t ) ) où le contrôle u est une fonction mesurable arbitraire de [ 0 , + ∞ [ dans { 0 , 1 } . Les conditions données ne nécessitent aucune intégration ou construction d’une fonction de Lyapunov...

Stability and gradient dynamical systems.

Jack K. Hale (2004)

Revista Matemática Complutense

The objective in these notes is to present an approach to dynamical systems in infinite dimensions. It does not seem reasonable to make a comparison of all of the orbits of the dynamics of two systems on non locally compact infinite dimensional spaces. Therefore, we choose to compare them on the set of globally defined bounded solutions. Fundamental problems are posed and several important results are stated when this set is compact. We then give results on the dynamical system which will ensure...

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