Configurations centrales en toile d'araignée

Ce mémoire est consacré à une classe remarquable de solutions du problème des N corps appelées configurations centrales. Ces configurations, notamment pour des corps coplanaires, sont étroitement liées aux solutions homographiques : en tout temps, la position des corps est obtenue par homothétie et/ou rotation de la position initiale. Notre objectif est d’étudier l’existence de configurations centrales en toile d’araignée données par n×ℓ masses disposées aux points d’intersection de n cercles concentriques avec ℓ demi-droite concourante, sous l’hypothèse que les ℓ masses du i ᶱ cercle sont égales à une constante positive mᵢ ; nous discutons également le cas où nous ajoutons une masse centrale m₀ au point de concours des ℓ demi-droites. Une première méthode analytique amène à l’existence de ces configurations centrales lorsque n = 2,3,4 et ℓ arbitraire pour n’importe quelles valeurs strictement positives de m₁, . . . , mₙ. Une seconde méthode analytique donne l’existence et l’unicité de telles configurations centrales lorsqu’on contraint ℓ à être égal à 2, . . . ,9 et n arbitraire pour n’importe quelles valeurs strictement positives de m₁, . . . , mₙ. De plus, nous étendons le résultat pour ℓ = 10, . . . ,18 en imposant m₁ ≥ · · · ≥ mₙ et en bornant la valeur de n dans chaque cas. En outre, pour ces deux méthodes analytiques, nous démontrons que les résultats restent vrais pour les configurations a N = n × ℓ + 1 corps, c’est-à-dire lorsqu’on ajoute une masse strictement positive au centre de masse. Enfin, nous donnons un algorithme permettant de prouver rigoureusement l’existence et l’unicité locale d’une telle configuration centrale avec un choix arbitraire de n, ℓ et m₁, . . . , mₙ. L’algorithme est applique à tous les n ≤ 100 et toutes les valeurs paires ℓ ≤ 200 lorsque m₁ = . . . = mₙ = 1/ℓ. Ceci est suffisant pour montrer l’existence des configurations centrales en toile d’araignée pour tout n ≤ 100, ℓ ≤ 200 pair et telles que m₁ = . . . = mₙ pour n’importe quelle valeur strictement positive.This thesis is dedicated to the study of a specific class of solutions for the N-body problem called central configurations. These configurations, especially in the planar case, are closely related to homographic solutions: at any time, the position of the bodies can be obtained by a rotation and/or a rescaling of the initial position. Our aim is to prove the existence of spiderweb central configurations given by n × ℓ masses located at the intersection of n concentric circles with ℓ concurrent half-lines, under the hypothesis that the ℓ masses on the i-th circle are equal to a positive constant mᵢ ; we also discuss the case where we add a central mass m₀ located at the intersection of the ℓ halflines. A first analytical method leads to the existence of these central configurations when n = 2,3,4 and ℓ arbitrary for any strictly positive values of m₁, . . . ,mₙ. A second analytical method yields the existence and uniqueness of such central configurations when we restrict ℓ to be equal to 2, . . . ,9 and n arbitrary for any strictly positive values of m₁, . . . , mₙ. In addition, we extend the result for ℓ = 10, . . . ,18 by requiring m₁ ≥ · · · ≥ mₙ and bounding the value of n in each case. Furthermore, for these two analytical methods, we demonstrate that the results hold for spiderweb configurations with N = n × ℓ + 1 bodies, that is when we add a strictly positive mass at the center of mass. Finally, we give an algorithm providing a rigorous proof of the existence and local uniqueness of such a central configuration with an arbitrary choice of n, ℓ and m₁, . . . , mₙ. The algorithm is applied to all n ≤ 100 and all even values ℓ ≤ 200 when m₁ = . . . = mₙ = 1/ℓ. This is enough to show the existence of spiderweb central configurations for all n ≤ 100, ℓ ≤ 200 even and such that m₁ = . . . = mₙ for any value strictly positive

Numerical computation of transverse homoclinic orbits for periodic solutions of delay differential equations

We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the \emph{Poincar\'{e} scenario}. The strategy is geometric in nature, and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fr\'{e}chet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold and the BVP is solved using a discrete time dynamical system approach (defined via the \emph{method of steps}) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey-Glass systems

Constructive proofs for localized radial solutions of semilinear elliptic systems on Rd\mathbb{R}^d

Ground state solutions of elliptic problems have been analyzed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localized solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localized radially symmetric solutions of elliptic systems on $\mathbb{R}^d$. Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov-Perron operator with a Newton-Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localized radial solutions of the cubic Klein-Gordon equation on $\mathbb{R}^3$, the Swift-Hohenberg equation on $\mathbb{R}^2$, and a three-component FitzHugh-Nagumo system on $\mathbb{R}^2$. These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs

Periodic orbits in Hořava–Lifshitz cosmologies

We consider spatially homogeneous Hořava–Lifshitz models that perturb General Relativity (GR) by a parameter v∈(0,1) such that GR occurs at v=1/2. We describe the dynamics for the extremal case v=0, which possess the usual Bianchi hierarchy: type I (Kasner circle of equilibria), type II (heteroclinics that induce the Kasner map) and type VI0,VII0 (further heteroclinics). For type VIII and IX, we use a computer-assisted approach to prove the existence of periodic orbits which are far from the Mixmaster attractor. Therefore we obtain a new behaviour which is not described by the BKL picture of bouncing Kasner-like states

Periodic orbits in Ho\v{r}ava-Lifshitz cosmologies

We consider spatially homogeneous Ho\v{r}ava-Lifshitz (HL) models that perturb General Relativity (GR) by a parameter $v\in (0,1)$ such that GR occurs at $v=1/2$. We describe the dynamics for the extremal case $v=0$, which possess the usual Bianchi hierarchy: type I (Kasner circle of equilibria), type II (heteroclinics that induce the Kasner map) and type $\mathrm{VI_0},\mathrm{VII_0}$ (further heteroclinics). For type VIII and IX, we prove the existence of periodic orbits which are far from the Mixmaster attractor, and thereby yield a new behaviour which is not described by the BKL picture.Comment: 19 pages, 7 figure