Relevant bibliographies by topics / KAM and Nekhoroshev theory

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Academic literature on the topic 'KAM and Nekhoroshev theory'

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Published: 4 June 2021
Last updated: 31 January 2023

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Journal articles on the topic "KAM and Nekhoroshev theory"

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Benettin, Giancarlo, Francesco Fassò, and Massimiliano Guzzo. "Nekhoroshev-Stability ofL4andL5in the Spatial Restricted Problem." International Astronomical Union Colloquium 172 (1999): 445–46. http://dx.doi.org/10.1017/s0252921100073097.

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The Lagrangian equilateral pointsL4andL5of the restricted circular three-body problem are elliptic for all values of the reduced massμbelow Routh’s critical massμR≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekhoroshev-stability’: denoting byda convenient distance from the equilibrium point, one asks whetherfor any small єe > 0, with positiveaandb. Until recently this problem, as more generally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus onμ(see e.g .Giorgilli, 1989). Our aim was instead considering all values ofμup toμR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian exists and satisfies a ‘quasi-convexity’ assumption (Fassòet al, 1998; Guzzoet al, 1998; Niedermann, 1998).
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Guzzo, Massimiliano. "Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems." International Astronomical Union Colloquium 172 (1999): 443–44. http://dx.doi.org/10.1017/s0252921100073085.

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Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.
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Wiggins, S., and A. M. Mancho. "Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori." Nonlinear Processes in Geophysics 21, no. 1 (February 4, 2014): 165–85. http://dx.doi.org/10.5194/npg-21-165-2014.

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Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.
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Li, Yong, and Yingfei Yi. "Nekhoroshev and KAM Stabilities in Generalized Hamiltonian Systems." Journal of Dynamics and Differential Equations 18, no. 3 (July 15, 2006): 577–614. http://dx.doi.org/10.1007/s10884-006-9025-2.

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Bounemoura, Abed, and Stéphane Fischler. "A Diophantine duality applied to the KAM and Nekhoroshev theorems." Mathematische Zeitschrift 275, no. 3-4 (May 22, 2013): 1135–67. http://dx.doi.org/10.1007/s00209-013-1174-5.

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Bounemoura, Abed, and Laurent Niederman. "Generic Nekhoroshev theory without small divisors." Annales de l’institut Fourier 62, no. 1 (2012): 277–324. http://dx.doi.org/10.5802/aif.2706.

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Moan, Per Christian. "On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth." Nonlinearity 17, no. 1 (September 29, 2003): 67–83. http://dx.doi.org/10.1088/0951-7715/17/1/005.

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MacKay, R. S., and I. C. Percival. "Converse KAM: Theory and practice." Communications in Mathematical Physics 98, no. 4 (December 1985): 469–512. http://dx.doi.org/10.1007/bf01209326.

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Salamon, Dietmar, and Eduard Zehnder. "KAM theory in configuration space." Commentarii Mathematici Helvetici 64, no. 1 (December 1989): 84–132. http://dx.doi.org/10.1007/bf02564665.

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Delshams, Amadeu, and Pere Gutiérrez. "Effective Stability and KAM Theory." Journal of Differential Equations 128, no. 2 (July 1996): 415–90. http://dx.doi.org/10.1006/jdeq.1996.0102.

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Dissertations / Theses on the topic "KAM and Nekhoroshev theory"

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Saha, Prasenjit. "A perturbation method from KAM theory with applications to stellar and asteroidal motion." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.315813.

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Pageault, Pierre. "Fonctions de Lyapunov : une approche KAM faible." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2011. http://tel.archives-ouvertes.fr/tel-00678325.

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Cette thèse est divisée en trois parties. Dans une première partie, on donne une description nouvelle des points récurrents par chaînes d'un système dynamique comme ensemble d'Aubry projeté d'une barrière ultramétrique. Cette approche permet de munir l'ensemble des composantes transitives par chaînes d'une structure d'espace ultramétrique expliquant leur topologie totalement discontinue, et de retrouver un théorème célèbre de Charles Conley concernant l'existence de fonctions de Lyapunov décroissant strictement le long des orbites non-récurrentes par chaînes. Dans une deuxième partie, on développe une théorie d'Aubry-Mather pour les homéomorphismes d'un espace métrique compact. On introduit dans ce cadre un ensemble d'Aubry métrique, puis topologique, ainsi qu'un ensemble de Mañé. Ces notions, plus fines que la récurrence par chaînes, permettent de mieux comprendre les fonctions de Lyapunov d'un tel système dynamique. Dans une dernière partie, on montre un résultat général de densité de certains contre-exemples au théorème de Sard pour lesquels l'ensemble des points critiques est un arc topologique et on donne des applications dynamiques de ce résultat. Celles-ci sont liées à des problèmes d'unicité, à constantes près, des solutions KAM faibles (ou solutions de viscosité) de certaines équations d'Hamilton-Jacobi.
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Viveros, Rogel Jorge. "An extension of KAM theory to quasi-periodic breather solutions in Hamiltonian lattice systems." Diss., Georgia Institute of Technology, 2007. http://hdl.handle.net/1853/19869.

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We prove the existence and linear stability of quasi-periodic breather solutions in a 1d Hamiltonian lattice of identical, weakly-coupled, anharmonic oscillators with general on-site potentials and under the effect of long-ranged interaction, via de KAM technique. We prove the persistence of finite-dimensional tori which correspond in the uncoupled limit to N arbitrary lattice sites initially excited. The frequencies of the invariant tori of the perturbed system are only slightly deformed from the frequencies of the unperturbed tori.
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Mandorino, Vito. "Théorie KAM faible et instabilité pour familles d'hamiltoniens." Phd thesis, Université Paris Dauphine - Paris IX, 2013. http://tel.archives-ouvertes.fr/tel-00867687.

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Dans cette thèse nous étudions la dynamique engendrée par une famille de flots Hamiltoniens. Un tel système dynamique à plusieurs générateurs est aussi appelé 'polysystème'. Motivés par des questions liées au phénomène de la diffusion d'Arnold, notre objectif est de construire des trajectoires du polysystème qui relient deux régions lointaines de l'espace des phases. La thèse est divisée en trois parties.Dans la première partie, nous considérons le polysystème engendré par les flots discrétisés d'une famille d'Hamiltoniens Tonelli. En utilisant une approche variationnelle issue de la théorie KAM faible, nous donnons des conditions suffisantes pour l'existence des trajectoires souhaitées.Dans la deuxième partie, nous traitons le cas d'un polysystème engendré par un couple de flots Hamiltoniens à temps continu, dont l'étude rentre dans le cadre de la théorie géométrique du contrôle. Dans ce contexte, nous montrons dans certains cas la transitivité d'un polysystème générique, à l'aide du théorème de transversalité de Thom.La dernière partie de la thèse est dédiée à obtenir une nouvelle version du théorème de transversalité de Thom s'exprimant en termes d'ensembles rectifiables de codimension positive. Dans cette partie il n'est pas question de polysystèmes, ni d'Hamiltoniens. Néanmoins, les résultats obtenus ici sont utilisés dans la deuxième partie de la thèse
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Haro, Àlex. "The Primitive Function of an Exact Symplectomorphism. Variational principles, Converse KAM Theory and the problems of determination and interpolation." Doctoral thesis, Universitat de Barcelona, 1998. http://hdl.handle.net/10803/2116.

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We have divided this thesis in four parts:

a) PART I: Exact symplectic geometry (introduction of the problems). This part contains the basic tools of symplectic geometry and outlines the four subjects that we have study along the thesis: the determination problem, the interpolation problem, the variational problem and the breakdown problem.

b) PART II: On the standard symplectic manifold (analytical part). We recall the necessary tools to work on R(d) x R(d). That is we perform a coordinate treatment of the results. First of all we relate different kinds of generating functions to the primitive function and later we solve formally the determination problem. Then we introduce different variational principles: for fixed points, periodic orbits and orbital segments. Their invariance under certain kind of transformations of phase space is proved, and we interpret physically such results. Finally we give the basic properties of invariant exact Lagrangian graphs obtaining at last that if our graph is minimizing then its orbits are minimizing.

c) PART III: On the cotangent bundle (geometrical part). The first three chapters are similar to the three previous ones with the difference that we do an intrinsic treatment of the results by considering any cotangent bundle. The fourth chapter in this part deals with the solution of the interpolation problem given in analytic set up.

d) PART IV: Converse KAM theory (numerical part). The last part deals with the applications to converse Kolmogorv-Arnold-Moser (KAM) theory. First of all we give a small list of different examples that we shall study later. Then we generalize converse KAM theory and we related it to the Lipschitz theory by Birkhoff and Herman. Then we perform our variational Greene method and apply it to different examples. Also we study numerically the Aubry-Mather sets in higher dimensions. After this we apply our methods to the rotational standard map that is a symplectic skew product. Then we give some ideas about the geometrical obstructions for existence of invariant tori showing them with a simple example. We also find some known Birkhoff normal forms using our methods. Finally we explain briefly how our theory can be used for arbitrary Lagrangian foliations.
La present memòria es troba dividida en quatre parts ben diferenciades. La primera conté les eines bàsiques de la geometria simplèctica i planteja els quatre problemes que tractarem al llarg de la memòria: el problema de determinació, el problema d'interpolació, el problema variacional i el problema del trencament de tors invariants.

La segona part tracta sobre la varietat simpléctica estàndard, i vindria a ser la part analítica. Aquí hem treballat a R(d) x R(d), és a dir hem fet un tractament coordenat dels resultats. Primer relacionem les funcions generatrius amb la funció primitiva i després resolem formalment el problema de determinación. Tot seguit tractem diferents principis variacionals per als punts fixos per a les òrbites periòdiques i per als segments orbitals. La seva invariància respecte a certs tipus de transformacions de l'espai de fase és demostrada donant una interpretació física. Finalment donem les propietats bàsiques dels grafs Lagrangians invariants, especialment aquella que diu que les òrbites sobre un graf minimitzant són minimitzants.

La tercera part abraça el tema del fibrat cotangent, la part geométrica de l'obra. Els tres primers capítols segueixen més o menys la línia dels tres precedents amb la diferéncia fonamental que aquí considerem qualsevol fibrat cotangent. Fem llavors un tractament intrínsec. El quart capítol d'aquesta part està dedicat a resoldre el problema d'interpolació en el cas analític.

La quarta i darrera part (que vindria a ser la secció numèrica de la tesi), tracta de les aplicacions a la teoria Kolmogorv, Arnold i Moser (KAM) inversa o del trencament dels tors invariants. Primer donem una llista d'exemples que utilitzarem més endavant. Després generalitzem la teoria KAM inversa i la relacionem amb la teoria Lipschitziana de Birkhoff i Herman. Llavors implementem el nostre criteri de Greene variacional i l'apliquem a diferents exemples. També estudiem els equivalents dels conjunts d'Aubry-Mather en dimensió alta (bé = 4). Després apliquem aquesta metodologia a l'aplicació estàndard rotacional (3D), indicant abans la teoria necessària. Llavors donem algunes idees de com generalitzar els criteris obstruccionals a dimensions altes hi ho mostrem amb un petit exemple. Finalment retrobem algunes formes normals de Birkhoff utilitzant la nostra metodologia basada en la funcióprimitiva i expliquem una mica com es podria considerar la nostra teoria tenint en compte foliacions Lagrangianes arbitràries.
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Fontanari, Daniele. "Quantum manifestations of the adiabatic chaos of perturbed susperintegrable Hamiltonian systems." Thesis, Littoral, 2013. http://www.theses.fr/2013DUNK0356/document.

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Dans cette thèse nous étudions un système quantique, obtenu comme un analogue d'un système classique superintégrable perturbé au moyen de la quantification géométrique. Notre objectif est de mettre en évidence la présence des phénomènes analogues à ceux qui caractérisent la superintégrabilité classique, notamment la coexistence des mouvements réguliers et chaotiques liés aux effets des résonances ainsi que la régularité du régime non-résonant. L'analyse est effectuée par l'étude des distributions du Husimi des états quantiques sélectionnés, avec une attention particulière aux états stationnaires et à l'évolution des états cohérents. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Bien que cette thèse devrait être considérée comme une étude préliminaire, dont l'objectif est de créer le socle des études futures, nos résultats donnent des indications intéressantes sur la dynamique quantique. Par exemple, il est démontré comment les résonancees classiques exercent une influence considérable sur le spectre du système quantique et comment il est possible, dans le comportement quantique, de trouver une trace de l'invariant adiabatique dans le régime de résonance
The abundance, among physical models, of perturbations of superintegrable Hamiltonian systems makes the understanding of their long-term dynamics an important research topic. While from the classical standpoint the situation, at least in many important cases, is well understood through the use of Nekhoroshev stability theorem and of the adiabatic invariants theory, in the quantum framework there is, on the contrary, a lack of precise results. The purpose of this thesis is to study a perturbed superintegrable quantum system, obtained from a classical counterpart by means of geometric quantization, in order to highlight the presence of indicators of superintegrability analogues to the ones that characterize the classical system, such as the coexistence of regular motions with chaotic one, due to the effects of resonances, opposed to the regularity in the non resonant regime. The analysis is carried out by studying the Husimi distributions of chosen quantum states, with particular emphasis on stationary states and evolved coherent states. The computation are performed using both numerical methods and perturbative schemes. Although this should be considered a preliminary work, the purpose of which is to lay the fundations for future investigations, the results obtained here give interesting insights into quantum dynamics. For instance, it is shown how classical resonances exert a considerable influence on the spectrum of the quantum system and how it is possible, in the quantum behaviour, to find a trace of the classical adiabatic invariance in the resonance regime
L'abbondanza, fra i modelli fisici, di perturbazioni di sistemi Hamiltoniani superintegrabili rende la comprensione della loro dinamica per tempi lunghi un importante argomento diricerca. Mentre dal punto di vista classico la situazione, perlomeno in molti case importanti, è ben compresa grazie all'uso del teorema di stabilità di Nekhoroshev e della teoria degli invariantiadiabatici, nel caso quantistico vi è, al contrario, una mancanza di risultati precisi. L'obiettivo di questa tesi è di studiare un sistema superintegrabile quantistico, ottenuto partendo da un corrispettivo classico tramite quantizzazione geometrica, al fine di evidenziare la presenza di indicatori di supertintegrabilità analoghi a quelliche caratterizzano il sistema classico, come la coesistenza di moti regolari e caotici, dovuta all'effetto delle risonanze, in contrapposizione con la regolarità nel regime non risonante. L'analisi è condotta studiando le distribuzioni di Husimi di stati quantistici scelti, con particolare enfasi posta sugli stati stazionari e sugli stati coerenti evoluti. I calcoli sono effettuati sia utilizzando tecniche numeriche che schemi perturbativi. Pur essendo da considerardi questo un lavoro preliminare, il cui compito è di porre le fondamenta per analisi future, i risultati qui ottenuti offrono interessanti spunti sulla dinamica quantistica. Per esempio è mostrato come le risonanze classiche abbiano un chiaro effeto sullo spettro del sistema quantistico, ed inoltre comesia possibile trovare una traccia, nel comportamento quantistico, dell'invarianza adiabatica classica nel regime risonante
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Masoero, Marco. "On the long time behavior of potential MFG." Thesis, Paris Sciences et Lettres (ComUE), 2019. http://www.theses.fr/2019PSLED057.

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Cette thèse porte sur l’étude du comportement en temps long des jeux à champ moyen (MFG) potentiels, indépendamment de la convexité du problème de minimisation associé. Pour le système hamiltonien de dimension finie, des problèmes de même nature ont été traités par la théorie KAM faible. Nous transposons de nombreux résultats de cette théorie dans le contexte des jeux à champ moyen potentiels. Tout d'abord, nous caractérisons par approximation ergodique la valeur limite associée aux systèmes MFG à horizon fini. Nous fournissons des exemples explicites dans lesquels cette valeur est strictement supérieure au niveau d’énergie des solutions stationnaires du système MFG ergodique. Cela implique que les trajectoires optimales des systèmes MFG à horizon fini ne peuvent pas converger vers des configurations stationnaires. Ensuite, nous prouvons la convergence du problème de minimisation associé à MFG à horizon fini vers une solution de l’équation Hamilton-Jacobi critique dans l’espace de mesures de probabilité. De plus, nous montrons une limite de champ moyen pour la constante ergodique associée à l’équation Hamilton-Jacobi de dimension finie correspondante. Dans la dernière partie, nous caractérisons la limite du problème de minimisation à horizon infini que nous avons utilisé pour l'approximation ergodique dans la première partie du manuscrit
The purpose of this thesis is to shed some light on the long time behavior of potential Mean Field Games (MFG), regardless of the convexity of the minimization problem associated. For finite dimensional Hamiltonian systems, problems of the same nature have been addressed through the so-called weak KAM theory. We transpose many results of this theory in the infinite dimensional context of potential MFG. First, we characterize through an ergodic approximation the limit value associated to time dependent MFG systems. We provide explicit examples where this value is strictly greater than the energy level of stationary solutions of the ergodic MFG system. This implies that optimal trajectories of time dependent MFG systems cannot converge to stationary configurations. Then, we prove the convergence of the minimization problem associated to time dependent MFGs to a solution of the critical Hamilton-Jacobi equation in the space of probability measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding finite dimensional Hamilton-Jacobi equation. In the last part we characterize the limit of the infinite horizon discounted minimization problem that we use for the ergodic approximation in the first part of the manuscript
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Castan, Thibaut. "Stability in the plane planetary three-body problem." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066062/document.

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Arnold a démontré l'existence de solutions quasipériodiques dans le problème planétaire à trois corps plan, sous réserve que la masse de deux des corps, les planètes, soit petite par rapport à celle du troisième, le Soleil. Cette condition de petitesse dépend de façon cachée de la largeur d'analyticité de l'hamiltonien du problème, dans des coordonnées transcendantes. Hénon ex- plicita un rapport de masses minimal nécessaire à l'application du théorème de Arnold. L'objectif de cette thèse sera de donner une condition suffisante sur les rapports de masses. Une première partie de mon travail consiste à estimer cette largeur d'analyticité, ce qui passe par l'étude précise de l'équation de Kepler dans le complexe, ainsi que celle des singularités complexes de la fonction perturbatrice. Une deuxième partie consiste à mettre l'hamiltonien sous forme normale, dans l'optique d'une application du théorème KAM (du nom de Kolmogorov-Arnold-Moser). Il est nécessaire d'étudier le hamiltonien séculaire pour le mettre sous une forme normale adéquate. On peut alors quantifier la non-dégénérescence de l'hamiltonien séculaire, ainsi qu'estimer la perturbation. Enfin, il faut démontrer une version quantitative fine du théorème KAM, inspirée de Pöschel, avec des constantes explicites. A l'issue de ce travail, il est montré que le théorème KAM peut être appliqué pour des rapports de masses entre planètes et étoile de l'ordre de 10^(-85)
Arnold showed the existence of quasi-periodic solutions in the plane planetary three-body prob- lem, provided that the mass of two of the bodies, the planets, is small compared to the mass of the third one, the Sun. This smallness condition depends in a sensitive way on the analyticity widths of the Hamiltonian of the three-body problem, expressed with the help of some tran- scendental coordinates. Hénon gave a minimal ratio of masses necessary to the application of Arnold’s theorem. The main objective of this thesis is to determine a sufficient condition on this ratio. A first part of this work consists in estimating these analyticity widths, which requires a precise study of the complex Kepler equation, as well as the complex singularities of the disturb- ing function. A second part consists in reworking the Hamiltonian to put it under normal form, in order to apply the KAM theorem (KAM standing for Kolmogorov-Arnold-Moser). In this aim, it is essential to work with the secular Hamiltonian to put it under a suitable normal form. We can then quantify the non-degeneracy of the secular Hamiltonian, as well as estimate the perturbation. Finally, it is necessary to derive a quantitative version of the KAM theorem, in order to identify the hypotheses necessary for its application to the plane three-body problem. After this work, it is shown that the KAM theorem can be applied for a ratio of masses that is close to 10^(−85) between the planets and the star
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Valvo, Lorenzo. "Hamiltonian perturbation theory on a poisson algebra : application to a throbbing top and to magnetically confined particles." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0498.

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La théorie de perturbation Hamiltonienne de la mécanique classique est basé sur la structure d'algébre de Lie. Mais on trouve des structures de Lie dans tout les systèmes dit ``de Poisson''. Dans la première partie de cette thèse, on propose une approche purement algébrique à la théorie classique des perturbations, qui s'applique donc à tout les système de Poisson. Dans cette méthode, introduit en [Vittot, 2004] une transformation (de Lie) permet de diviser la perturbation en un terme préservant le flot non perturbé, et une correction quadratique.Dans l'exemple d'une Toupie Pulsante (un corps rigide non autonome) symétrique et périodique on montre que notre théorème s'applique et reproduit le théorème de KAM de la mécanique classique. Puis on considere une Toupie non symétrique avec moments d'inertie qui presentent des fluctuations quelconques: dans ce cas, on etudie sous quelles conditions les trajectoires du systeme sont proches de celle du système statique.Dans la troisième partie de ce mémoire, on étudie la dynamique d'une particule chargée dans un champ électromagnétique donné arbitraire. Par par la théorie des perturbations on peut réduire la dimensionnalité de la dynamique, ou étudier la rétroaction de la particule sur le champ. Cependant, fournir une description du flot non perturbé est une tâche redoutable, liée à la question de longue date de la théorie du centre-guide en physique des plasmas. Nous dérivons les équations du mouvement et leur structure de Poisson dans une nuovelle description relativiste et non perturbative de cette théorie
The Hamiltonian perturbation theory of classical mechanics is based on the underlying Lie algebraic structure. But Lie structures are met in a wider class of dynamical systems, called Poisson systems. In the first part of this thesis, we propose a purely algebraic approach to classical perturbation theory to extend its scope to any Poisson system. In this method, introduced in [Vittot, 2004], a (Lie) transform allows to split the perturbation into a term reserving the unperturbed flow, and a smaller correction, quadratic in the original perturbation strength.The second part of the dissertation is about the dynamics of a non-autonomous Top. We consider first a symmetric Top with periodically dependent moments of inertia; in this case, our theorem applies and reproduces the KAM theorem of classical mechanics. Then we switch to a non symmetric Top with non-periodically fluctuating moments of inertia: in this case we study for which conditions the static trajectories give a good approximation to those of the non-autonomous system.In the third part of this work we study the dynamics of a magnetically confined particle. By perturbation theory one may reduce the dimensionality of the dynamics, or study the retroaction of the particle on the field. However, providing an efficient description of the unperturbed flow is a formidable task, related to the long-standing issue of Guiding Centre Theory in plasma physics. Recently a novel relativistic and non-perturbative approach to Guiding Centre theory has been proposed [Di Troia, 2018]. We derive the equations of motion and their Poisson structure in this description
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10

Reinol, Alisson de Carvalho [UNESP]. "Integrabilidade e dinâmica global de sistema diferenciais polinomiais definidos em R³ com superfícies algébricas invariantes de graus 1 e 2." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/151140.

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Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Neste trabalho, consideramos aspectos algébricos e dinâmicos de alguns problemas envolvendo superfícies algébricas invariantes em sistemas diferenciais polinomiais definidos em R³. Determinamos o número máximo de planos invariantes que um sistema diferencial quadrático pode ter e estudamos a realização e integrabilidade de tais sistemas. Fornecemos a forma normal para sistemas diferenciais com quádricas invariantes e estudamos de forma mais detalhada a dinâmica e integrabilidade de sistemas diferenciais quadráticos com um paraboloide elíptico como superfície algébrica invariante. Por fim, estudamos as consequências dinâmicas ao se perturbar um sistema diferencial, cujo espaço de fase é folheado por superfícies algébricas invariantes. Para tal, consideramos o sistema diferencial quadrático conhecido como sistema Sprott A, que depende de um parâmetro real a e apresenta comportamento caótico mesmo sem ter pontos de equilíbrio, tendo, assim, um hidden attractor para valores adequados do parâmetro a. Provamos que, para a=0, o espaço de fase desse sistema é folheado por esferas concêntricas invariantes. Utilizando a Teoria do Averaging e o Teorema KAM (Kolmogorov-Arnold-Moser), provamos que, para a>0 suficientemente pequeno, uma órbita periódica orbitalmente estável emerge de um equilíbrio do tipo zero-Hopf não isolado localizado na origem e que formam-se toros invariantes em torno desta órbita periódica. Concluímos que a ocorrência de tais fatos tem um papel importante na formação do hidden attractor.
In this work, we consider algebraic and dynamical aspects of some problems involving invariant algebraic surfaces in polynomial differential systems defined in R³. We determine the maximum number of invariant planes that a quadratic differential system can have and we study the realization and integrability of such systems. We provide the normal form for differential systems having an invariant quadric and we study in more detail the dynamics and integrability of quadratic differential systems having an elliptic paraboloid as invariant algebraic surface. Finally, we study the dynamic consequences of perturbing differential system whose phase space is foliated by invariant algebraic surfaces. For this we consider the quadratic differential system known as Sprott A system, which depends on one real parameter a and presents chaotic behavior even without having any equilibrium point, thus having a hidden attractor for suitable values of parameter a. We prove that, for a=0, the phase space of this system is foliated by invariant concentric spheres. By using the Averaging Theory and the KAM (Kolmogorov-Arnold-Moser) Theorem, we prove that, for a>0 sufficiently small, an orbitally stable periodic orbit emerges from a zero-Hopf nonisolated equilibrium point located at the origin and that invariant tori are formed around this periodic orbit. We conclude that the occurrence of these facts has an important role in the formation of the hidden attractor.
FAPESP: 2013/26602-7
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Books on the topic "KAM and Nekhoroshev theory"

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Giancarlo, Benettin, Henrard J, Kuksin Sergej B. 1955-, Giorgilli Antonio, Centro internazionale matematico estivo, and European Mathematical Society, eds. Hamiltonian dynamics theory and applications: Lectures given at the C.I.M.E.-E.M.S. Summer School, held in Cetraro, Italy, July 1-10, 1999. Berlin: Springer, 2005.

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Kappeler, Thomas. KdV & KAM. Berlin: Springer, 2003.

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Jürgen, Pöschel, ed. KdV & KAM. Berlin: Springer, 2003.

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Lazutkin, Vladimir F. KAM theory andsemiclassical approximations to eigenfunctions. Berlin: Springer-Verlag, 1993.

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Luo, Albert C. J., and Valentin Afraimovich, eds. Hamiltonian Chaos Beyond the KAM Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-12718-2.

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Lazutkin, V. F. KAM theory and semiclassical approximations to eigenfunctions. Berlin: Springer-Verlag, 1993.

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Lazutkin, Vladimir F. KAM Theory and Semiclassical Approximations to Eigenfunctions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993.

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Lazutkin, Vladimir F. KAM Theory and Semiclassical Approximations to Eigenfunctions. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993. http://dx.doi.org/10.1007/978-3-642-76247-5.

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González-Enríquez, A. Singularity theory for non-twist KAM tori. Providence, Rhode Island: American Mathematical Society, 2013.

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Valentin, Afraimovich, and SpringerLink (Online service), eds. Hamiltonian Chaos Beyond the KAM Theory: Dedicated to George M. Zaslavsky (1935–2008). Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2011.

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Book chapters on the topic "KAM and Nekhoroshev theory"

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Niederman, Laurent. "Nekhoroshev Theory." In Mathematics of Complexity and Dynamical Systems, 1070–81. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-1806-1_62.

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Niederman, Laurent. "Nekhoroshev Theory." In Encyclopedia of Complexity and Systems Science, 5986–98. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-30440-3_352.

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Niederman, Laurent. "Nekhoroshev Theory." In Perturbation Theory, 291–305. New York, NY: Springer US, 2009. http://dx.doi.org/10.1007/978-1-0716-2621-4_352.

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Delshams, Amadeu, and Pere Gutiérrez. "Nekhoroshev and KAM Theorems Revisited via a Unified Approach." In Hamiltonian Mechanics, 299–306. Boston, MA: Springer US, 1994. http://dx.doi.org/10.1007/978-1-4899-0964-0_29.

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Broer, Henk, and Floris Takens. "On KAM theory." In Dynamical Systems and Chaos, 173–204. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-6870-8_5.

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Lazutkin, Vladimir F. "KAM Theorems." In KAM Theory and Semiclassical Approximations to Eigenfunctions, 121–59. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993. http://dx.doi.org/10.1007/978-3-642-76247-5_4.

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Guzzo, Massimiliano, Zoran Knežević, and Andrea Milani. "Probing the Nekhoroshev Stability of Asteroids." In Modern Celestial Mechanics: From Theory to Applications, 121–40. Dordrecht: Springer Netherlands, 2002. http://dx.doi.org/10.1007/978-94-017-2304-6_8.

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Meyer, Kenneth R., and Glen R. Hall. "Stability and KAM Theory." In Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 227–40. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-4073-8_9.

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Meyer, Kenneth R., and Daniel C. Offin. "Stability and KAM Theory." In Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 305–44. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-53691-0_12.

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Meyer, Kenneth, Glen Hall, and Dan Offin. "Stability and KAM Theory." In Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 329–54. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-09724-4_13.

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Conference papers on the topic "KAM and Nekhoroshev theory"

1

Manarvi, Abdul, and Troy Henderson. "Tracking GPS orbits using KAM theory." In 2017 IEEE Aerospace Conference. IEEE, 2017. http://dx.doi.org/10.1109/aero.2017.7943567.

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CARDIN, F. "FLUID DYNAMICAL FEATURES OF THE WEAK KAM THEORY." In Proceedings of the 14th Conference on WASCOM 2007. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812772350_0018.

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KALOSHIN, VADIM YU. "MATHER THEORY, WEAK KAM THEORY, AND VISCOSITY SOLUTIONS OF HAMILTON-JACOBI PDE'S." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0004.

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YUAN, XIAOPING. "RECENT PROGRESS ON NONLINEAR WAVE EQUATIONS VIA KAM THEORY." In Control Theory and Related Topics - In Memory of Professor Xunjing Li. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812790552_0027.

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González-Enríquez, A., and R. de la Llave. "Analytic approximations of geometric maps and applications to KAM theory." In Proceedings of the International Conference on SPT 2007. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812776174_0037.

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Yang, Yang, and Yanning Zheng. "A Review of Information Acquisition Based on Information Foraging Theory." In 2009 Second International Symposium on Knowledge Acquisition and Modeling. IEEE, 2009. http://dx.doi.org/10.1109/kam.2009.42.

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Zhong, Qi, Zheng Li, and Le Zhang. "Analysis of the Emergency Procurement Based on Evolutionary Game Theory." In 2009 Second International Symposium on Knowledge Acquisition and Modeling (KAM 2009). IEEE, 2009. http://dx.doi.org/10.1109/kam.2009.67.

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Chen, Jingpu. "Evolutionary Research on Financial Core Competence Based on Complex Systematic Theory." In 2009 Second International Symposium on Knowledge Acquisition and Modeling. IEEE, 2009. http://dx.doi.org/10.1109/kam.2009.133.

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Duan, Weihua, Xinhua Bi, and Yawei Wang. "The Trend Analysis of HRM Outsourcing Relationship Based on Game Theory." In 2009 Second International Symposium on Knowledge Acquisition and Modeling. IEEE, 2009. http://dx.doi.org/10.1109/kam.2009.301.

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Li Li, Qiu Meng, and Wu Bei. "Based on queuing theory to solve the optimization number of berth." In 2010 3rd International Symposium on Knowledge Acquisition and Modeling (KAM). IEEE, 2010. http://dx.doi.org/10.1109/kam.2010.5646271.

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