/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

For a convex, real analytic, e-close to integrable Hamiltonian system with n≥5 degrees of freedom, we construct an orbit exhibiting Arnold diffusion with the diffusion time bounded by $\exp(C\epsilon^{-\frac{1}{2(n-2)}})$
 . This upper bound of the diffusion time almost matches the lower bound of order $\exp(\epsilon ^{-\frac{1}{2(n-1)}})$
 predicted by the Nekhoroshev-type stability results. Our method is based on the variational approach of Bessi and Mather, and includes a new construction on the space of frequencies.

Speed of Arnold diffusion for analytic Hamiltonian systems

In the present paper we prove a strong form of Arnold diusion. Let T 2 be the two torus and B 2 be the unit ball around the origin in R 2 . Fix > 0. Our main result says that for a \generic" time-periodic perturbation of an integrable system of two degrees of freedom H0(p) +"H1(;p;t ); 2 T 2 ; p2 B 2 ; t2 T = R=Z; with a strictly convexH0, there exists a -dense orbit ( ;p ;t)(t) in T 2 B 2 T, namely, a -neighborhood of the orbit contains T 2 B 2 T. Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are usage of crumpled normally hyperbolic invariant cylinders from [13], ower and simple normally hyperbolic invariant manifolds from [47] as well as their kissing property at a strong double resonance. This allows us to build a \connected" net of 3-dimensional normally hyperbolic invariant manifolds. To construct diusing orbits along this net we

A strong form of Arnold diusion for two and a half degrees of freedom

In this article, we present a new approach of Nekhoroshev theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak which combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

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Generic Nekhoroshev theory without small divisors

For perturbations of integrable Hamiltonian systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is analytic. This fundamental result has been extended in two distinct directions. The first one is due to Niederman, who showed that under the analyticity assumption, the result holds true for a prevalent class of integrable systems which is much wider than the steep systems. The second one is due to Marco-Sauzin but it is limited to quasi-convex integrable systems, for which they showed exponential stability if the system is assumed to be only Gevrey regular. If the system is finitely differentiable, we showed polynomial stability, still in the quasi-convex case. The goal of this work is to generalize all these results in a unified way, by proving exponential or polynomial stability for Gevrey or finitely differentiable perturbations of prevalent integrable Hamiltonian systems.

https://hal.archives-ouvertes.fr/hal-00539595/document

Effective Stability for Gevrey and Finitely Differentiable Prevalent Hamiltonians

In this paper, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper lower bound on the time of Arnold diffusion which we believe to be optimal.

Abed Bounemoura

Papers

Generic Nekhoroshev theory without small divisors

Effective Stability for Gevrey and Finitely Differentiable Prevalent Hamiltonians

Generic Nekhoroshev theory without small divisors

Improved exponential stability for near-integrable quasi-convex Hamiltonians

Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians