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Johannes Gutenberg Universität Mainz
1.
Albrecht, Joachim.
Über die Existenz invarianter Tori in Hamiltonschen Systemen, die bis auf eine endlich oft differenzierbare Störung analytisch und integrabel sind.
Degree: 2005, Johannes Gutenberg Universität Mainz
URL: http://ubm.opus.hbz-nrw.de/volltexte/2005/830/ 
► Es wird die Existenz invarianter Tori in Hamiltonschen Systemen bewiesen, die bis auf eine 2n-mal stetig differenzierbare Störung analytisch und integrabel sind, wobei n die…
(more)
▼ Es wird die Existenz invarianter Tori in Hamiltonschen Systemen bewiesen, die bis auf eine 2n-mal stetig differenzierbare Störung analytisch und integrabel sind, wobei n die Anzahl der Freiheitsgrade bezeichnet. Dabei wird vorausgesetzt, dass die Stetigkeitsmodule der 2n-ten partiellen Ableitungen der Störung einer Endlichkeitsbedingung (Integralbedingung) genügen, welche die Hölderbedingung verallgemeinert. Bisher konnte die Existenz invarianter Tori nur unter der Voraussetzung bewiesen werden, dass die 2n-ten Ableitungen der Störung hölderstetig sind.
We prove the existence of invariant tori in Hamiltonian Systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom). It is assumed that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.
Subjects/Keywords: KAM-Theorie; KAM-Theory; Mathematics
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APA (6th Edition):
Albrecht, J. (2005). Ãœber die Existenz invarianter Tori in Hamiltonschen Systemen, die bis auf eine endlich oft differenzierbare Störung analytisch und integrabel sind. (Doctoral Dissertation). Johannes Gutenberg Universität Mainz. Retrieved from http://ubm.opus.hbz-nrw.de/volltexte/2005/830/
Chicago Manual of Style (16th Edition):
Albrecht, Joachim. “Ãœber die Existenz invarianter Tori in Hamiltonschen Systemen, die bis auf eine endlich oft differenzierbare Störung analytisch und integrabel sind.” 2005. Doctoral Dissertation, Johannes Gutenberg Universität Mainz. Accessed January 25, 2021.
http://ubm.opus.hbz-nrw.de/volltexte/2005/830/.
MLA Handbook (7th Edition):
Albrecht, Joachim. “Ãœber die Existenz invarianter Tori in Hamiltonschen Systemen, die bis auf eine endlich oft differenzierbare Störung analytisch und integrabel sind.” 2005. Web. 25 Jan 2021.
Vancouver:
Albrecht J. Ãœber die Existenz invarianter Tori in Hamiltonschen Systemen, die bis auf eine endlich oft differenzierbare Störung analytisch und integrabel sind. [Internet] [Doctoral dissertation]. Johannes Gutenberg Universität Mainz; 2005. [cited 2021 Jan 25].
Available from: http://ubm.opus.hbz-nrw.de/volltexte/2005/830/.
Council of Science Editors:
Albrecht J. Ãœber die Existenz invarianter Tori in Hamiltonschen Systemen, die bis auf eine endlich oft differenzierbare Störung analytisch und integrabel sind. [Doctoral Dissertation]. Johannes Gutenberg Universität Mainz; 2005. Available from: http://ubm.opus.hbz-nrw.de/volltexte/2005/830/
2.
Masoero, Marco.
On the long time behavior of potential MFG : Sur la convergence en temps long des jeux à champ moyen de type potentiel.
Degree: Docteur es, Sciences, 2019, Paris Sciences et Lettres (ComUE); École nationale supérieure des mines (Paris)
URL: http://www.theses.fr/2019PSLED057
► 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…
(more)
▼ 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.
Advisors/Committee Members: Cardaliaguet, Pierre (thesis director).
Subjects/Keywords: Jeux à champ moyen; Contrôl optimal; Théorie KAM faible; Mean Field Games; Optimal Control; Weak KAM theory; 519
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APA (6th Edition):
Masoero, M. (2019). On the long time behavior of potential MFG : Sur la convergence en temps long des jeux à champ moyen de type potentiel. (Doctoral Dissertation). Paris Sciences et Lettres (ComUE); École nationale supérieure des mines (Paris). Retrieved from http://www.theses.fr/2019PSLED057
Chicago Manual of Style (16th Edition):
Masoero, Marco. “On the long time behavior of potential MFG : Sur la convergence en temps long des jeux à champ moyen de type potentiel.” 2019. Doctoral Dissertation, Paris Sciences et Lettres (ComUE); École nationale supérieure des mines (Paris). Accessed January 25, 2021.
http://www.theses.fr/2019PSLED057.
MLA Handbook (7th Edition):
Masoero, Marco. “On the long time behavior of potential MFG : Sur la convergence en temps long des jeux à champ moyen de type potentiel.” 2019. Web. 25 Jan 2021.
Vancouver:
Masoero M. On the long time behavior of potential MFG : Sur la convergence en temps long des jeux à champ moyen de type potentiel. [Internet] [Doctoral dissertation]. Paris Sciences et Lettres (ComUE); École nationale supérieure des mines (Paris); 2019. [cited 2021 Jan 25].
Available from: http://www.theses.fr/2019PSLED057.
Council of Science Editors:
Masoero M. On the long time behavior of potential MFG : Sur la convergence en temps long des jeux à champ moyen de type potentiel. [Doctoral Dissertation]. Paris Sciences et Lettres (ComUE); École nationale supérieure des mines (Paris); 2019. Available from: http://www.theses.fr/2019PSLED057
University of Colorado
3.
Fox, Adam Merritt.
Destruction of Invariant Tori in Volume-Preserving Maps.
Degree: PhD, Applied Mathematics, 2013, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/36
► Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and KAM theory…
(more)
▼ Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and
KAM theory guarantees the persistence of some tori upon perturbation. When these tori have codimension-one they act as boundaries to transport, and therefore play a prominent role in the global stability of the system. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though
KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. This dissertation begins by extending Greene's residue criterion to three-dimensional, reversible, volume-preserving maps.
The application of Greene's residue criterion requires the repeated computation of periodic orbits, which is costly if the system is nonreversible. We describe a quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a torus and demonstrate how the growth of the Sobolev norm or singular values of this conjugacy can be used to predict criticality. We will then use this method to study both reversible and nonreversible volume-preserving maps in two and three dimensions. The near-critical conjugacies, and the gaps that form within them, will be explored in the context of Aubry-Mather and Anti-Integrability
theory, when applicable. This dissertation will conclude by exploring the locally and globally most robust tori in area-preserving maps.
Advisors/Committee Members: James D. Meiss, Juan Restrepo, Keith Julien, Elizabeth Bradley, James Curry.
Subjects/Keywords: KAM theory; Greene's residue criterion; near-critical conjugacies; Applied Mathematics
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APA (6th Edition):
Fox, A. M. (2013). Destruction of Invariant Tori in Volume-Preserving Maps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/36
Chicago Manual of Style (16th Edition):
Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 25, 2021.
https://scholar.colorado.edu/appm_gradetds/36.
MLA Handbook (7th Edition):
Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Web. 25 Jan 2021.
Vancouver:
Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 25].
Available from: https://scholar.colorado.edu/appm_gradetds/36.
Council of Science Editors:
Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/36
University of Texas – Austin
4.
Maciejewski, James Michael.
An application of KAM theory to a model for particle channelling in crystals and some related numerical simulations.
Degree: MA, Mathematics, 2010, University of Texas – Austin
URL: http://hdl.handle.net/2152/ETD-UT-2010-05-1092
► It has been seen in physical experiments as early as the 1960’s that when a positively charged particle is injected into a crystal in certain…
(more)
▼ It has been seen in physical experiments as early as the 1960’s that when a positively charged particle is injected into a crystal in certain directions and with sufficiently
high energy, the particle can penetrate into the crystalline structure to a depth which is
unexpectedly long. Such motions in the crystal are referred to as channelling trajectories.
Herein, we consider a Hamiltonian model for particle motion in a crystal. We then proceed
to show that the results of
KAM theory are applicable to the model, and moreover that these
result predict the existence of trajectories for the fired particle which do indeed penetrate
the crystal deeply. Finally, we present the results of two series of numerical simulations
which strongly suggest that this behaviour is observable in our model.
Advisors/Committee Members: Llave, Rafael de la (advisor), Bichteler, Klaus (committee member).
Subjects/Keywords: KAM theory; Particle channelling; Crystals
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APA (6th Edition):
Maciejewski, J. M. (2010). An application of KAM theory to a model for particle channelling in crystals and some related numerical simulations. (Masters Thesis). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/ETD-UT-2010-05-1092
Chicago Manual of Style (16th Edition):
Maciejewski, James Michael. “An application of KAM theory to a model for particle channelling in crystals and some related numerical simulations.” 2010. Masters Thesis, University of Texas – Austin. Accessed January 25, 2021.
http://hdl.handle.net/2152/ETD-UT-2010-05-1092.
MLA Handbook (7th Edition):
Maciejewski, James Michael. “An application of KAM theory to a model for particle channelling in crystals and some related numerical simulations.” 2010. Web. 25 Jan 2021.
Vancouver:
Maciejewski JM. An application of KAM theory to a model for particle channelling in crystals and some related numerical simulations. [Internet] [Masters thesis]. University of Texas – Austin; 2010. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/2152/ETD-UT-2010-05-1092.
Council of Science Editors:
Maciejewski JM. An application of KAM theory to a model for particle channelling in crystals and some related numerical simulations. [Masters Thesis]. University of Texas – Austin; 2010. Available from: http://hdl.handle.net/2152/ETD-UT-2010-05-1092
Australian National University
5.
Gomes, Sean P.
Quantum ergodicity in mixed and KAM Hamiltonian systems
.
Degree: 2017, Australian National University
URL: http://hdl.handle.net/1885/154331
► In this thesis, we investigate quantum ergodicity for two classes of Hamiltonian systems satisfying intermediate dynamical hypotheses between the well understood extremes of ergodic flow…
(more)
▼ In this thesis, we investigate quantum ergodicity for two classes
of Hamiltonian systems
satisfying intermediate dynamical hypotheses between the well
understood extremes of
ergodic flow and quantum completely integrable flow. These two
classes are mixed Hamiltonian systems and KAM Hamiltonian
systems.
Hamiltonian systems with mixed phase space decompose into
finitely many invariant
subsets, only some of which are of ergodic character. It has been
conjectured by Percival that the eigenfunctions of the
quantisation of this system decompose into associated
families of analogous character. The first project in this thesis
proves a weak form of this
conjecture for a class of dynamical billiards, namely the
mushroom billiards of Bunimovich
for a full measure subset of a shape parameter t in (0,2].
KAM Hamiltonian systems arise as perturbations of completely
integrable Hamiltonian
systems. The dynamics of these systems are well understood and
have near-integrable
character. The classical-quantum correspondence suggests that the
quantisation of KAM
systems will not have quantum ergodic character. The second
project in this thesis proves
an initial negative quantum ergodicity result for a class of
positive Gevrey perturbations
of a Gevrey Hamiltonian that satisfy a mild slow torus condition.
Subjects/Keywords: quantum ergodicity;
microlocal analysis;
semiclassical analysis;
partial differential equations;
spectral theory;
KAM Hamiltonian systems
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APA (6th Edition):
Gomes, S. P. (2017). Quantum ergodicity in mixed and KAM Hamiltonian systems
. (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/154331
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Gomes, Sean P. “Quantum ergodicity in mixed and KAM Hamiltonian systems
.” 2017. Thesis, Australian National University. Accessed January 25, 2021.
http://hdl.handle.net/1885/154331.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Gomes, Sean P. “Quantum ergodicity in mixed and KAM Hamiltonian systems
.” 2017. Web. 25 Jan 2021.
Vancouver:
Gomes SP. Quantum ergodicity in mixed and KAM Hamiltonian systems
. [Internet] [Thesis]. Australian National University; 2017. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1885/154331.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Gomes SP. Quantum ergodicity in mixed and KAM Hamiltonian systems
. [Thesis]. Australian National University; 2017. Available from: http://hdl.handle.net/1885/154331
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
6.
Pageault, Pierre.
Fonctions de Lyapunov : une approche KAM faible : Lyapunov functions : a weak KAM approach.
Degree: Docteur es, Mathématiques, 2011, Lyon, École normale supérieure
URL: http://www.theses.fr/2011ENSL0654
► 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…
(more)
▼ 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.
This thesis is divided into three parts. In the first part, we give a new description of chain-recurrence using an ultrametric barrier. This barrier allows to endow the space of chain-transitive components with an ultrametric structure, explaining its topology and leading to the famous result of Charles Conley about Lyapunov function decreasing along non chain-recurrent orbits. Most of the results, first given in the setting of a continuous map on a compact metric space are then generalised to multivalued map on arbitrary separable metric spaces. In the second part, we develop an Aubry-Mather theory for a homeomorphism on a compact metric space. In this setting, we introduce metric and topological Aubry set and Mañé set, allowing a better understanding of Lyapunov functions arising in such a dynamical system. In the last part, we prove a general density result for some counterexamples of Sard's theorem for which the set of critical points is a topological arc and we give applications to dynamics.
Advisors/Committee Members: Fathi, Albert (thesis director).
Subjects/Keywords: Fonctions de Lyapunov; Systèmes dynamiques; Récurrence par chaînes; Théorème de Sard; Théorie KAM faible; Lyapunov function; Dynamical systems; Chain-recurrence; Sard's theorem; Weak KAM theory
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APA ·
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APA (6th Edition):
Pageault, P. (2011). Fonctions de Lyapunov : une approche KAM faible : Lyapunov functions : a weak KAM approach. (Doctoral Dissertation). Lyon, École normale supérieure. Retrieved from http://www.theses.fr/2011ENSL0654
Chicago Manual of Style (16th Edition):
Pageault, Pierre. “Fonctions de Lyapunov : une approche KAM faible : Lyapunov functions : a weak KAM approach.” 2011. Doctoral Dissertation, Lyon, École normale supérieure. Accessed January 25, 2021.
http://www.theses.fr/2011ENSL0654.
MLA Handbook (7th Edition):
Pageault, Pierre. “Fonctions de Lyapunov : une approche KAM faible : Lyapunov functions : a weak KAM approach.” 2011. Web. 25 Jan 2021.
Vancouver:
Pageault P. Fonctions de Lyapunov : une approche KAM faible : Lyapunov functions : a weak KAM approach. [Internet] [Doctoral dissertation]. Lyon, École normale supérieure; 2011. [cited 2021 Jan 25].
Available from: http://www.theses.fr/2011ENSL0654.
Council of Science Editors:
Pageault P. Fonctions de Lyapunov : une approche KAM faible : Lyapunov functions : a weak KAM approach. [Doctoral Dissertation]. Lyon, École normale supérieure; 2011. Available from: http://www.theses.fr/2011ENSL0654
7.
Castan, Thibaut.
Stability in the plane planetary three-body problem : Stabilité dans le problème à trois corps planétaire plan.
Degree: Docteur es, Mathématiques appliquées, 2017, Université Pierre et Marie Curie – Paris VI
URL: http://www.theses.fr/2017PA066062
► 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…
(more)
▼ 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.
Advisors/Committee Members: Féjoz, Jacques (thesis director).
Subjects/Keywords: Problème à trois corps; Théorie des perturbations; Théorème KAM; Systèmes hamiltoniens; Systèmes dynamiques; Géométrie symplectique; Three-body problem; Perturbation theory; KAM theorem; 519.6
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APA (6th Edition):
Castan, T. (2017). Stability in the plane planetary three-body problem : Stabilité dans le problème à trois corps planétaire plan. (Doctoral Dissertation). Université Pierre et Marie Curie – Paris VI. Retrieved from http://www.theses.fr/2017PA066062
Chicago Manual of Style (16th Edition):
Castan, Thibaut. “Stability in the plane planetary three-body problem : Stabilité dans le problème à trois corps planétaire plan.” 2017. Doctoral Dissertation, Université Pierre et Marie Curie – Paris VI. Accessed January 25, 2021.
http://www.theses.fr/2017PA066062.
MLA Handbook (7th Edition):
Castan, Thibaut. “Stability in the plane planetary three-body problem : Stabilité dans le problème à trois corps planétaire plan.” 2017. Web. 25 Jan 2021.
Vancouver:
Castan T. Stability in the plane planetary three-body problem : Stabilité dans le problème à trois corps planétaire plan. [Internet] [Doctoral dissertation]. Université Pierre et Marie Curie – Paris VI; 2017. [cited 2021 Jan 25].
Available from: http://www.theses.fr/2017PA066062.
Council of Science Editors:
Castan T. Stability in the plane planetary three-body problem : Stabilité dans le problème à trois corps planétaire plan. [Doctoral Dissertation]. Université Pierre et Marie Curie – Paris VI; 2017. Available from: http://www.theses.fr/2017PA066062
Penn State University
8.
Chen, Dong.
On some problems in Lagrangian Dynamics and Finsler Geometry.
Degree: 2017, Penn State University
URL: https://submit-etda.libraries.psu.edu/catalog/14421dxc360
► The purpose of this dissertation is to present several applications of enveloping functions and dual lens maps to geometry and dynamical systems. In Chapter 1…
(more)
▼ The purpose of this dissertation is to present several applications of enveloping functions and dual lens maps to geometry and dynamical systems. In Chapter 1 we have a brief review on basic notions and
theory we need to understand the main results. In Chapter 2 we prove that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points. The major tool is enveloping functions.
In Chapter 3 we introduce the dual lens map technique developed by Burago and Ivanov. It derives from enveloping functions and symplectic geometry. We then show how this technique is used to perturb the geodesic flows of flat Finsler tori.
In Chapter 4 we show how dual lens map can be used in
KAM theory. The celebrated
KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as
KAM tori. What happens outside
KAM tori draws a lot of attention. We show two types of Lagrangian perturbations of the geodesic flow on flat Finsler tori. The perturbations are C^∞ small but the resulting flows has a positive measure of trajectories with positive Lyapunov exponent. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy outside some
KAM tori and it gives positive answer to a question asked by Kolmogorov.
Advisors/Committee Members: Dmitri Yu Burago, Dissertation Advisor/Co-Advisor, Dmitri Yu Burago, Committee Chair/Co-Chair, Federico Juan Rodriguez Hertz, Committee Member, Mark Levi, Committee Member, Runze Li, Outside Member.
Subjects/Keywords: KAM theory; Finsler metric; duel lens map Hamiltonian flow; perturbation; metric entropy; duel lens map; Hamiltonian flow; perturbation; metric entropy
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APA (6th Edition):
Chen, D. (2017). On some problems in Lagrangian Dynamics and Finsler Geometry. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/14421dxc360
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Chen, Dong. “On some problems in Lagrangian Dynamics and Finsler Geometry.” 2017. Thesis, Penn State University. Accessed January 25, 2021.
https://submit-etda.libraries.psu.edu/catalog/14421dxc360.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Chen, Dong. “On some problems in Lagrangian Dynamics and Finsler Geometry.” 2017. Web. 25 Jan 2021.
Vancouver:
Chen D. On some problems in Lagrangian Dynamics and Finsler Geometry. [Internet] [Thesis]. Penn State University; 2017. [cited 2021 Jan 25].
Available from: https://submit-etda.libraries.psu.edu/catalog/14421dxc360.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Chen D. On some problems in Lagrangian Dynamics and Finsler Geometry. [Thesis]. Penn State University; 2017. Available from: https://submit-etda.libraries.psu.edu/catalog/14421dxc360
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
9.
Liu, Jian-Long.
Preservation of Periodicity in Variational Integrators.
Degree: MS, Mathematics, 2015, San Jose State University
URL: https://doi.org/10.31979/etd.c4z9-y9yp
;
https://scholarworks.sjsu.edu/etd_theses/4596
► Classical numerical integrators do not preserve symplecticity, a structure inherent in Hamiltonian systems. Thus, the trajectories they produce cannot be expected to possess the…
(more)
▼ Classical numerical integrators do not preserve symplecticity, a structure inherent in Hamiltonian systems. Thus, the trajectories they produce cannot be expected to possess the same qualitative behavior observed in the original system. Pooling recent results from O'Neale and West, we explore a particular class of numerical integrators, the variational integrator, that preserves one aspect of the range of behavior present in Hamiltonian systems, the periodicity of trajectories. We first establish the prerequisites and some key concepts from Hamiltonian systems, particularly symplecticity and action-angle coordinates. Through perturbation theory and its complications manifested in small divisor problems, we motivate the necessity for KAM theory. With O'Neale's KAM-type theorem, we observe the preservation of periodicity by symplectic one-step methods. Lastly, we show that the variational integrator introduced by West possesses the defining characteristics of symplectic one-step methods, and therefore also preserves periodicity of the original trajectories.
Subjects/Keywords: hamiltonian system; kam theory; periodicity; perturbation theory; symplectic integrator; variational integrator
…Kolmogorov-Arnold-Moser (KAM) theory from
classical perturbation theory, with a sketch of… …CHAPTER 3
KAM THEORY
In this chapter, we attempt to motivate and explain the progression of the… …initial
development of the subject of KAM theory through perturbation theory. We then state the… …original
coordinates.
3.1
Perturbation Theory
KAM theory has its roots in classical perturbation… …KAM theory through the
problems of small divisors.
In perturbation theory, given a Liouville…
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APA (6th Edition):
Liu, J. (2015). Preservation of Periodicity in Variational Integrators. (Masters Thesis). San Jose State University. Retrieved from https://doi.org/10.31979/etd.c4z9-y9yp ; https://scholarworks.sjsu.edu/etd_theses/4596
Chicago Manual of Style (16th Edition):
Liu, Jian-Long. “Preservation of Periodicity in Variational Integrators.” 2015. Masters Thesis, San Jose State University. Accessed January 25, 2021.
https://doi.org/10.31979/etd.c4z9-y9yp ; https://scholarworks.sjsu.edu/etd_theses/4596.
MLA Handbook (7th Edition):
Liu, Jian-Long. “Preservation of Periodicity in Variational Integrators.” 2015. Web. 25 Jan 2021.
Vancouver:
Liu J. Preservation of Periodicity in Variational Integrators. [Internet] [Masters thesis]. San Jose State University; 2015. [cited 2021 Jan 25].
Available from: https://doi.org/10.31979/etd.c4z9-y9yp ; https://scholarworks.sjsu.edu/etd_theses/4596.
Council of Science Editors:
Liu J. Preservation of Periodicity in Variational Integrators. [Masters Thesis]. San Jose State University; 2015. Available from: https://doi.org/10.31979/etd.c4z9-y9yp ; https://scholarworks.sjsu.edu/etd_theses/4596
Georgia Tech
10.
Viveros Rogel, Jorge.
An extension of KAM theory to quasi-periodic breather solutions in Hamiltonian lattice systems.
Degree: PhD, Mathematics, 2007, Georgia Tech
URL: http://hdl.handle.net/1853/19869
► 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…
(more)
▼ 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.
Advisors/Committee Members: Yi, Yingfei (Committee Chair), Bellissard, Jean (Committee Member), Dieci, Luca (Committee Member), Verriest, Erik I. (Committee Member), Weiss, Howard (Committee Member).
Subjects/Keywords: KAM theory; Hamiltonian; Lattice; Quasi-periodic breathers; Oscillations; Hamiltonian systems; Energy transfer; Lattice theory
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APA ·
Chicago ·
MLA ·
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APA (6th Edition):
Viveros Rogel, J. (2007). An extension of KAM theory to quasi-periodic breather solutions in Hamiltonian lattice systems. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/19869
Chicago Manual of Style (16th Edition):
Viveros Rogel, Jorge. “An extension of KAM theory to quasi-periodic breather solutions in Hamiltonian lattice systems.” 2007. Doctoral Dissertation, Georgia Tech. Accessed January 25, 2021.
http://hdl.handle.net/1853/19869.
MLA Handbook (7th Edition):
Viveros Rogel, Jorge. “An extension of KAM theory to quasi-periodic breather solutions in Hamiltonian lattice systems.” 2007. Web. 25 Jan 2021.
Vancouver:
Viveros Rogel J. An extension of KAM theory to quasi-periodic breather solutions in Hamiltonian lattice systems. [Internet] [Doctoral dissertation]. Georgia Tech; 2007. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1853/19869.
Council of Science Editors:
Viveros Rogel J. An extension of KAM theory to quasi-periodic breather solutions in Hamiltonian lattice systems. [Doctoral Dissertation]. Georgia Tech; 2007. Available from: http://hdl.handle.net/1853/19869
Universiteit Utrecht
11.
Rink, B.W.
Geometry and dynamics in Hamiltonian lattices.
Degree: 2003, Universiteit Utrecht
URL: http://dspace.library.uu.nl:8080/handle/1874/884
► E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a…
(more)
▼ E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a one-dimensional continuum. The model consisted of a discrete number of equal point masses that interact with their nearest neighbours only. On the basis of statistical mechanics, they expected that when the interparticle forces were anharmonic, the lattice would reach a thermal equilibrium. This means that averaged over time, its initial energy would be equipartitioned among all its Fourier modes. The famous computer experiment that they performed in Los Alamos in 1955, was intended to investigate in what manner and at what time-scale the equipartitioning would take place. The result was astonishing: the lattice did not come close to thermal equilibrium, but behaved more or less quasi-periodically. Only when the initial energy was larger than a certain threshold, did the lattice seem to `thermalise'. This paradox is nowadays known as the `Fermi-Pasta-Ulam problem'.
The observations of Fermi, Pasta and Ulam were a great impulse for nonlinear dynamics. One possible explanation for the quasi-periodic behaviour of the FPU system, is based on the Kolmogorov-Arnol'd-Moser theorem, which states that most of the invariant Lagrangean tori of a Liouville integrable Hamiltonian system survive under small perturbations. It is required though for the KAM theorem that the integrable system satisfies a nondegeneracy condition. Unfortunately, it has for a long time been unclear how the Fermi-Pasta-Ulam lattice can be viewed as a perturbation of a nondegenerate integrable system.
Nishida in 1971 and Sanders in 1977, investigated a Birkhoff normal form for the FPU lattice. Under the assumption of a rather strong nonresonance condition on the linear frequencies ωk = 2 \sin (kπ/n) of the lattice, they showed that this Birkhoff normal form is integrable and nondegenerate. This means that the KAM theorem can indeed be applied. The problem is of course that their required nonresonance condition is usually not met. This leaves a large gap in the proofs. The new idea in this thesis is to incorporate the discrete symmetries of the lattice in the argument. It turns out that this enables us to show that the nonresonance condition of Nishida and Sanders is not needed: every resonance is overruled by a symmetry. Hence the Birkhoff normal form is integrable and this proves the applicability of the KAM theorem.
Moreover, much attention is paid to the analysis of exact and approximate solutions of the lattice equations. New exact solutions and invariant manifolds are found in the fixed point sets of the symmetries of the lattice. Also, the Birkhoff normal reveals approximate integrals and solutions in the low energy domain of the phase space. The analysis makes use of invariant theory and singular reduction. One of the conclusions is that the lattice with an even number of particles contains travelling wave solutions that change their direction. Moreover the integrable normal form contains…
Subjects/Keywords: Wiskunde en Informatica; Fermi-Pasta-Ulam chain; Birkhoff normal forms; symmetry; resonance; dynamics; invariant manifolds; singular reduction; monodromy; KAM theory
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APA ·
Chicago ·
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APA (6th Edition):
Rink, B. W. (2003). Geometry and dynamics in Hamiltonian lattices. (Doctoral Dissertation). Universiteit Utrecht. Retrieved from http://dspace.library.uu.nl:8080/handle/1874/884
Chicago Manual of Style (16th Edition):
Rink, B W. “Geometry and dynamics in Hamiltonian lattices.” 2003. Doctoral Dissertation, Universiteit Utrecht. Accessed January 25, 2021.
http://dspace.library.uu.nl:8080/handle/1874/884.
MLA Handbook (7th Edition):
Rink, B W. “Geometry and dynamics in Hamiltonian lattices.” 2003. Web. 25 Jan 2021.
Vancouver:
Rink BW. Geometry and dynamics in Hamiltonian lattices. [Internet] [Doctoral dissertation]. Universiteit Utrecht; 2003. [cited 2021 Jan 25].
Available from: http://dspace.library.uu.nl:8080/handle/1874/884.
Council of Science Editors:
Rink BW. Geometry and dynamics in Hamiltonian lattices. [Doctoral Dissertation]. Universiteit Utrecht; 2003. Available from: http://dspace.library.uu.nl:8080/handle/1874/884
12.
Viana Camejo, Mikel.
Results on invariant whiskered tori for fibered holomorphic maps and on compensated domains.
Degree: PhD, Mathematics, 2018, Georgia Tech
URL: http://hdl.handle.net/1853/59176
We present a very general theory that includes results on the persistence of quasi-periodic orbits of systems subject to quasi-periodic perturbations.
Advisors/Committee Members: Loss, Michael (committee member), Zeng, Chongchun (committee member), Jorba, Angel (committee member), Bonetto, Federico (committee member).Subjects/Keywords: Quasi-periodic dynamics; KAM theory; Lower dimensional elliptic tori; Skew-products; Compensated domains
…50s and 60s (the KAM
theory). The systematic numerical treatment is more recent… …x29;
where ∆c0 := U0−1 ψ0−1 ∆c0 . It is standard in KAM theory to argue that the term
eR U0… …tori for fibered
holomorphic maps
In this paper we present a very general theory that… …kx − ykB . This condition is very important for the function theory in
D. This is joint… …1
CHAPTER I
INTRODUCTION
The goal of this paper is to present a very general theory of…
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APA ·
Chicago ·
MLA ·
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Export
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APA (6th Edition):
Viana Camejo, M. (2018). Results on invariant whiskered tori for fibered holomorphic maps and on compensated domains. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/59176
Chicago Manual of Style (16th Edition):
Viana Camejo, Mikel. “Results on invariant whiskered tori for fibered holomorphic maps and on compensated domains.” 2018. Doctoral Dissertation, Georgia Tech. Accessed January 25, 2021.
http://hdl.handle.net/1853/59176.
MLA Handbook (7th Edition):
Viana Camejo, Mikel. “Results on invariant whiskered tori for fibered holomorphic maps and on compensated domains.” 2018. Web. 25 Jan 2021.
Vancouver:
Viana Camejo M. Results on invariant whiskered tori for fibered holomorphic maps and on compensated domains. [Internet] [Doctoral dissertation]. Georgia Tech; 2018. [cited 2021 Jan 25].
Available from: http://hdl.handle.net/1853/59176.
Council of Science Editors:
Viana Camejo M. Results on invariant whiskered tori for fibered holomorphic maps and on compensated domains. [Doctoral Dissertation]. Georgia Tech; 2018. Available from: http://hdl.handle.net/1853/59176
13.
Rink, B.W.
Geometry and dynamics in Hamiltonian lattices.
Degree: 2003, University Utrecht
URL: https://dspace.library.uu.nl/handle/1874/884
;
URN:NBN:NL:UI:10-1874-884
;
1874/884
;
URN:NBN:NL:UI:10-1874-884
;
https://dspace.library.uu.nl/handle/1874/884
► E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a…
(more)
▼ E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a one-dimensional continuum. The model consisted of a discrete number of equal point masses that interact with their nearest neighbours only. On the basis of statistical mechanics, they expected that when the interparticle forces were anharmonic, the lattice would reach a thermal equilibrium. This means that averaged over time, its initial energy would be equipartitioned among all its Fourier modes. The famous computer experiment that they performed in Los Alamos in 1955, was intended to investigate in what manner and at what time-scale the equipartitioning would take place. The result was astonishing: the lattice did not come close to thermal equilibrium, but behaved more or less quasi-periodically. Only when the initial energy was larger than a certain threshold, did the lattice seem to `thermalise'. This paradox is nowadays known as the `Fermi-Pasta-Ulam problem'.
The observations of Fermi, Pasta and Ulam were a great impulse for nonlinear dynamics. One possible explanation for the quasi-periodic behaviour of the FPU system, is based on the Kolmogorov-Arnol'd-Moser theorem, which states that most of the invariant Lagrangean tori of a Liouville integrable Hamiltonian system survive under small perturbations. It is required though for the KAM theorem that the integrable system satisfies a nondegeneracy condition. Unfortunately, it has for a long time been unclear how the Fermi-Pasta-Ulam lattice can be viewed as a perturbation of a nondegenerate integrable system.
Nishida in 1971 and Sanders in 1977, investigated a Birkhoff normal form for the FPU lattice. Under the assumption of a rather strong nonresonance condition on the linear frequencies ωk = 2 \sin (kπ/n) of the lattice, they showed that this Birkhoff normal form is integrable and nondegenerate. This means that the KAM theorem can indeed be applied. The problem is of course that their required nonresonance condition is usually not met. This leaves a large gap in the proofs. The new idea in this thesis is to incorporate the discrete symmetries of the lattice in the argument. It turns out that this enables us to show that the nonresonance condition of Nishida and Sanders is not needed: every resonance is overruled by a symmetry. Hence the Birkhoff normal form is integrable and this proves the applicability of the KAM theorem.
Moreover, much attention is paid to the analysis of exact and approximate solutions of the lattice equations. New exact solutions and invariant manifolds are found in the fixed point sets of the symmetries of the lattice. Also, the Birkhoff normal reveals approximate integrals and solutions in the low energy domain of the phase space. The analysis makes use of invariant theory and singular reduction. One of the conclusions is that the lattice with an even number of particles contains travelling wave solutions that change their direction. Moreover the integrable normal form contains…
Subjects/Keywords: Fermi-Pasta-Ulam chain; Birkhoff normal forms; symmetry; resonance; dynamics; invariant manifolds; singular reduction; monodromy; KAM theory
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❌
APA ·
Chicago ·
MLA ·
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CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Rink, B. W. (2003). Geometry and dynamics in Hamiltonian lattices. (Doctoral Dissertation). University Utrecht. Retrieved from https://dspace.library.uu.nl/handle/1874/884 ; URN:NBN:NL:UI:10-1874-884 ; 1874/884 ; URN:NBN:NL:UI:10-1874-884 ; https://dspace.library.uu.nl/handle/1874/884
Chicago Manual of Style (16th Edition):
Rink, B W. “Geometry and dynamics in Hamiltonian lattices.” 2003. Doctoral Dissertation, University Utrecht. Accessed January 25, 2021.
https://dspace.library.uu.nl/handle/1874/884 ; URN:NBN:NL:UI:10-1874-884 ; 1874/884 ; URN:NBN:NL:UI:10-1874-884 ; https://dspace.library.uu.nl/handle/1874/884.
MLA Handbook (7th Edition):
Rink, B W. “Geometry and dynamics in Hamiltonian lattices.” 2003. Web. 25 Jan 2021.
Vancouver:
Rink BW. Geometry and dynamics in Hamiltonian lattices. [Internet] [Doctoral dissertation]. University Utrecht; 2003. [cited 2021 Jan 25].
Available from: https://dspace.library.uu.nl/handle/1874/884 ; URN:NBN:NL:UI:10-1874-884 ; 1874/884 ; URN:NBN:NL:UI:10-1874-884 ; https://dspace.library.uu.nl/handle/1874/884.
Council of Science Editors:
Rink BW. Geometry and dynamics in Hamiltonian lattices. [Doctoral Dissertation]. University Utrecht; 2003. Available from: https://dspace.library.uu.nl/handle/1874/884 ; URN:NBN:NL:UI:10-1874-884 ; 1874/884 ; URN:NBN:NL:UI:10-1874-884 ; https://dspace.library.uu.nl/handle/1874/884
University of Notre Dame
14.
Qun Ma.
Novel Multiscale Algorithms for Molecular
Dynamics</h1>.
Degree: Computer Science and Engineering, 2003, University of Notre Dame
URL: https://curate.nd.edu/show/4q77fq99533
► In post-genomic computational biology and bioinformatics, long simulations of the dynamics of molecular systems, particularly biological molecules such as proteins and DNA, require advances…
(more)
▼ In post-genomic computational biology and
bioinformatics, long simulations of the dynamics of molecular
systems, particularly biological molecules such as proteins and
DNA, require advances in time stepping computational methods. The
most severe problem of these algorithms is instability. The
objective of this dissertation is to present original work in
constructing multiscale multiple time stepping (MTS) algorithms for
molecular dynamics (MD) that allow large time steps. First, through
nonlinear stability analysis and numerical experiments, we reveal
that MTS integrators such as Impulse suffer nonlinear overheating
when Δ t = T/3 or possibly Δ t = T/4 when
constant–energy MD simulations are attempted, where Δ t is
the longest step size and T is the shortest period of the modes
in the system. Second, we present Targeted MOLLY â„¢, a new
multiscale integrator for MD simulations. TM combines an efficient
implementation of B-spline MOLLY exploiting analytical Hessians of
energies and a self–consistent dissipative leapfrog integrator.
Results show that TM allows very large time steps for slow forces
(and thus multiscale) for the numerically challenging flexible
TIP3P water systems (Jorgensen, {it et al.} J. Chem. Phys., vol 79,
pp 926–935, 1983) while still computing the dynamical and
structural properties accurately. Finally, we show yet another new
MOLLY integrator, the Backward Euler (BE) MOLLY in which hydrogen
bond forces can easily be included in the averaging and thus
stability might be further improved.
Advisors/Committee Members: Jesus A Izaguirre, Committee Chair, Menelaos I Karavelas, Committee Member, Ryan K Roeder, Committee Member, Gregory Madey, Committee Member, J Daniel Gezelter, Committee Member.
Subjects/Keywords: KAM theory; mollified Impulse method; nonlinear instability; targeted Langevin stabilization; long molecular dynamics simulations; multiple time stepping; Verlet-I/r-RESPA/Impulse
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APA ·
Chicago ·
MLA ·
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CSE |
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APA (6th Edition):
Ma, Q. (2003). Novel Multiscale Algorithms for Molecular
Dynamics</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/4q77fq99533
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Ma, Qun. “Novel Multiscale Algorithms for Molecular
Dynamics</h1>.” 2003. Thesis, University of Notre Dame. Accessed January 25, 2021.
https://curate.nd.edu/show/4q77fq99533.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Ma, Qun. “Novel Multiscale Algorithms for Molecular
Dynamics</h1>.” 2003. Web. 25 Jan 2021.
Vancouver:
Ma Q. Novel Multiscale Algorithms for Molecular
Dynamics</h1>. [Internet] [Thesis]. University of Notre Dame; 2003. [cited 2021 Jan 25].
Available from: https://curate.nd.edu/show/4q77fq99533.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Ma Q. Novel Multiscale Algorithms for Molecular
Dynamics</h1>. [Thesis]. University of Notre Dame; 2003. Available from: https://curate.nd.edu/show/4q77fq99533
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
15.
Mandorino, Vito.
Théorie KAM faible et instabilité pour familles d'hamiltoniens : Weak KAM theory and instability for families of Hamiltonians.
Degree: Docteur es, Mathématiques appliquées, 2013, Paris 9
URL: http://www.theses.fr/2013PA090003
► 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’.…
(more)
▼ 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
In this thesis we study the dynamics generated by a family of Hamiltonian flows. Such a dynamical system with several generators is also called ‘polysystem’.Motivated by some questions related to the phenomenon of Arnold diffusion, our aim is to construct trajectories of the polysystem which connect two far-apart regions of the phase space.The thesis is divided into three parts.In the first part, we consider the polysystem generated by the time-onemaps of a family of Tonelli Hamiltonians. By using a variational approach falling within the framework of weak KAM theory, we give sufficient conditions for the existence of the desired trajectories.In the second part, we address the case of a polysystem generated by twocontinuous-time Hamiltonian flows. This problem fits into the framework of geometriccontrol theory. In this context, we show in some cases the transitivity of a generic polysystem, by means of Thom’s transversality theorem.The third and last part of the thesis is devoted to the proof of a newversion of Thom’s transversality theorem, formulated in terms of rectifiable sets of positive codimension. Neither polysystems nor Hamiltonians are explicitly involved in this part. However, the results obtained here are used in the second part of the thesis.
Advisors/Committee Members: Bernard, Patrick (thesis director).
Subjects/Keywords: Dynamique hamiltonienne et lagrangienne; Théorie KAM faible; Diffusion d’Arnold; Polysystème; Semi-groupe de Lax-Oleinik; Ensembles d’Aubry et Mañé; Propriétés génériques; Théorie géométrique du contrôle; Ensemble atteignable; Théorème de transversalité de Thom; Ensemble rectifiable; Hamiltonian and Lagrangian dynamics; Weak KAM theory; Arnold diffusion; Polysystem; Lax-Oleinik semigroup; Aubry and Mañé sets; Generic properties; Geometric control theory; Reachable set; Thom’s transversality theorem; Rectifiable set
…approach of Mather and
Fathi’s weak KAM theory has been fruitful, especially in the framework of… …Kam theory, for which we refer to [Fat]. The
ideas will be close to those in… …issue de la théorie KAM faible.
Dans la Partie 2 nous considérons le cas d’un polysystème… …Polysystèmes hamiltoniens à temps discret (approche
avec la théorie KAM faible)
Dans cette… …Notons que les hamiltoniens Tonelli sont les hamiltoniens standard de la théorie KAM faible…
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APA ·
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APA (6th Edition):
Mandorino, V. (2013). Théorie KAM faible et instabilité pour familles d'hamiltoniens : Weak KAM theory and instability for families of Hamiltonians. (Doctoral Dissertation). Paris 9. Retrieved from http://www.theses.fr/2013PA090003
Chicago Manual of Style (16th Edition):
Mandorino, Vito. “Théorie KAM faible et instabilité pour familles d'hamiltoniens : Weak KAM theory and instability for families of Hamiltonians.” 2013. Doctoral Dissertation, Paris 9. Accessed January 25, 2021.
http://www.theses.fr/2013PA090003.
MLA Handbook (7th Edition):
Mandorino, Vito. “Théorie KAM faible et instabilité pour familles d'hamiltoniens : Weak KAM theory and instability for families of Hamiltonians.” 2013. Web. 25 Jan 2021.
Vancouver:
Mandorino V. Théorie KAM faible et instabilité pour familles d'hamiltoniens : Weak KAM theory and instability for families of Hamiltonians. [Internet] [Doctoral dissertation]. Paris 9; 2013. [cited 2021 Jan 25].
Available from: http://www.theses.fr/2013PA090003.
Council of Science Editors:
Mandorino V. Théorie KAM faible et instabilité pour familles d'hamiltoniens : Weak KAM theory and instability for families of Hamiltonians. [Doctoral Dissertation]. Paris 9; 2013. Available from: http://www.theses.fr/2013PA090003
. 
Open Access Theses and Dissertations
