21 research outputs found
Spatial discretizations of generic dynamical systems
How is it possible to read the dynamical properties (ie when the time goes to
infinity) of a system on numerical simulations? To try to answer this question,
we study in this manuscript a model reflecting what happens when the orbits of
a discrete time system (for example an homeomorphism) are computed
numerically . The computer working in finite numerical precision, it will
replace by a spacial discretization of , denoted by (where the
order of discretization stands for the numerical accuracy). In particular,
we will be interested in the dynamical behaviour of the finite maps for a
generic system and going to infinity, where generic will be taken in
the sense of Baire (mainly among sets of homeomorphisms or
-diffeomorphisms).
The first part of this manuscript is devoted to the study of the dynamics of
the discretizations , when is a generic conservative/dissipative
homeomorphism of a compact manifold. We show that it would be mistaken to try
to recover the dynamics of from that of a single discretization : its
dynamics strongly depends on the order . To detect some dynamical features
of , we have to consider all the discretizations when goes through
.
The second part deals with the linear case, which plays an important role in
the study of -generic diffeomorphisms, discussed in the third part of this
manuscript. Under these assumptions, we obtain results similar to those
established in the first part, though weaker and harder to prove.Comment: 322 pages. This is an improved version of the thesis of the author
(among others, the introduction and conclusion have been translated into
English). In particular, it contains works already published on arXiv.
Comments welcome
Cram\'er distance and discretizations of circle expanding maps I: theory
This paper is aimed to study the ergodic short-term behaviour of
discretizations of circle expanding maps. More precisely, we prove some
asymptotics of the distance between the -th iterate of Lebesgue measure by
the dynamics and the -th iterate of the uniform measure on the grid of
order by the discretization on this grid, when is fixed and the order
goes to infinity. This is done under some explicit genericity hypotheses on
the dynamics, and the distance between measures is measured by the mean of
\emph{Cram\'er} distance. The proof is based on a study of the corresponding
linearized problem, where the problem is translated into terms of
equirepartition on tori of dimension exponential in .
A numerical study associated to this work is presented in arXiv:2206.08000
[math.DS].Comment: 33 pages, 5 figure
Cram\'er distance and discretizations of circle expanding maps II: simulations
This paper presents some numerical experiments in relation with the
theoretical study of the ergodic short-term behaviour of discretizations of
expanding maps done in arXiv:2206.07991 [math.DS].
Our aim is to identify the phenomena driving the evolution of the Cram\'er
distance between the -th iterate of Lebesgue measure by the dynamics and
the -th iterate of the uniform measure on the grid of order by the
discretization on this grid. Based on numerical simulations we propose some
conjectures on the effects of numerical truncation from the ergodic viewpoint.Comment: 29 pages, 18 figure
Discrétisations spatiales de systèmes dynamiques génériques
How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations ? To try to answer this question, we study inthis thesis a model reflecting what happens when the orbits of a discrete time system f (for example an homeomorphism) are computed numerically. The computer working in finite numerical precision, it will replace f by a spacial discretization of f, denotedby f_N (where the order N of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or C^1-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations f_N, when f is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of f from that of a single discretization f_N : its dynamics strongly depends on the order N. To detect some dynamical features of f we have to consider all thediscretizations f_N when N goes through N.The second part deals with the linear case, which plays an important role in the study of C^1-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part,though weaker and harder to prove.Dans quelle mesure peut-on lire les propriétés dynamiques (quand le temps tend vers l’infini) d’un système sur des simulations numériques ? Pour tenter de répondre à cette question, on étudie dans cette thèse un modèle rendant compte de ce qui se passe lorsqu’on calcule numériquement les orbites d’un système à temps discret f (par exemple un homéomorphisme). L’ordinateur travaillant à précision numérique finie, il va remplacer f par une discrétisation spatiale de f, notée f_N (où l’ordre de la discrétisation N rend compte de la précision numérique). On s’intéresse en particulier au comportement dynamique des applications finies f_N pour un système f générique et pour l’ordre N tendant vers l’infini, où générique sera à prendre dans le sens de Baire (principalement parmi des ensembles d’homéomorphismes ou de C^1-difféomorphismes). La première partie de cette thèse est consacrée à l’étude de la dynamique des discrétisations f_N lorsque f est un homéomorphisme conservatif/dissipatif générique d’une variété compacte. L’étude montre qu’il est illusoire de vouloir retrouver la dynamique du système de départ f à partir de celle d’une seule discrétisation f_N : la dynamique de f_N dépend fortement de l’ordre N. Pour détecter certaines dynamiques de f il faut considérer l’ensemble des discrétisations f_N, lorsque N parcourt N.La seconde partie traite du cas linéaire, qui joue un rôle important dans l’étude du cas des C^1-difféomorphismes génériques, abordée dans la troisième partie de cette thèse. Sous ces hypothèses, on obtient des résultats similaires à ceux établis dans la première partie, bien que plus faibles et de preuves plus difficiles