Mather Sets Research Articles

Existence of solutions to contact mean-field games of first order

Abstract This paper deals with the existence of solutions of a class of contact mean-field game systems of first order consisting of a contact Hamilton-Jacobi equation and a continuity equation. Evans found a connection between Hamilton-Jacobi equations and continuity equations from the weak KAM point of view, where the coupling term is zero. Inspired by his work, we prove the main existence result by analyzing the properties of the Mather set for contact Hamiltonian systems.

The Aubry Set and Mather Set in the Embedded Contact Hamiltonian System

The Aubry Set and Mather Set in the Embedded Contact Hamiltonian System

Global Dynamics of a Vibro-Impact Energy Harvester

In this paper, we consider a two-sided vibro-impact energy harvester described as a forced cylindrical capsule inclined at a horizontal angle, and the motion of the ball inside the capsule follows from the impacts with the capsule ends and gravity. Two distinct cases of dynamical behavior are investigated: the nondissipative and dissipative cases, where the dissipation is given by a restitution coefficient of impacts. We show that the dynamics of the system are described by the use of a 2D implicit map written in terms of the variables’ energy and time when the ball leaves the moving capsule ends. More precisely, in the nondissipative case, we analytically show that this map is area-preserving and the existence of invariant curves for some rotation number with Markoff constant type is proved according to Moser’s twist theorem in high energy. The existence of invariant curves implies that the kinetic energy of the ball is always bounded, and hence, the structure of system is not destroyed by the impacts of the ball. Furthermore, by numerical analysis we also show that the dynamical behavior of this system is regular, mainly containing periodic points, invariant curves and Aubry–Mather sets. After introducing dissipation, the dissipation destroys the regular dynamical behavior of the nondissipative case, and a periodic point with low energy is generated.

The Existence of Aubry–Mather sets for the Fermi–Ulam Model

We consider the Fermi–Ulam model, which can be described as a particle moving freely between two vertical rigid walls; the left one being fixed, whereas the right one moves according to a regular periodic function. The particle is elastically reflected when hitting the walls. We show that the dynamics of the model can be described by an area-preserving monotone twist map. Thus, the Aubry–Mather sets exist for every rotation number in the rotation interval. Consequently, this gives a description of global dynamics behavior, particularly a large class of periodic and quasiperiodic orbits for the model.

On Finite Quotient Aubry set for Generic Geodesic Flows

We study the structure of the Mather and Aubry sets for the family of lagrangians given by the kinetic energy associated to a riemannian metric $ g$ on a closed manifold $ M$. In this case the Euler-Lagrange flow is the geodesic flow of $(M,g)$. We prove that there exists a residual subset $ \mathcal G$ of the set of all conformal metrics to $g$, such that, if $ \overline g \in \mathcal G$ then the corresponding geodesic flow has a finitely many ergodic c-minimizing measures, for each non-trivial cohomology class $ c \in H^1(M,\mathbb{R})$. This implies that, for any $ c \in H^1(M,\mathbb{R})$, the quotient Aubry set for the cohomology class c has a finite number of elements for this particular family of lagrangian systems.

A notion of Denjoy sub-system

We introduce a notion of Denjoy sub-system that generalizes that of the Aubry–Mather set. For such systems, we prove a result similar to Denjoy theorem (non-existence of C2 Denjoy sub-systems), and study their Lyapunov exponents.

Aubry–Mather Theory for Conformally Symplectic Systems

In this article we develop an analogue of Aubry-Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. Besides describing their structure and their dynamical significance, we shall analyze their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system.

Calibrated Configurations for Frenkel–Kontorova Type Models in Almost Periodic Environments

The Frenkel–Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We focus, in this context, on the existence of calibrated configurations (a notion stronger than the standard minimizing condition). In any dimension and for any continuous superlinear interaction energies, we exhibit a set, called projected Mather set, formed of environments that admit calibrated configurations. In the one-dimensional setting, we then give sufficient conditions on the family of interaction energies that guarantee the existence of calibrated configurations for every environment. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry–Mather theory and spaces of Delone sets.

Aubry–Mather sets for relativistic oscillators with anharmonic potentials

In this paper, by the Aubry–Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models $$\frac{d}{{dt}}\left( {\frac{{\dot x}}{{\sqrt {1 - {{\left| {\dot x} \right|}^2}} }}} \right) + {\left| x \right|^{\alpha - 1}}x = \;p\left( t \right),$$ where p(t) ∈ C0(R1) is 1-periodic and α > 0.

Remarks on the symplectic invariance of Aubry–Mather sets

In this note, we discuss and clarify some issues related to the generalization of Bernard's theorem on the symplectic invariance of Aubry, Mather and Mañé sets, to the cases of non-zero cohomology classes or non-exact symplectomorphisms, not necessarily homotopic to the identity.

Global Diffusion on a Tight Three-Sphere

We consider an integrable Hamiltonian system weakly coupled with a pendulum-type system. For each energy level within some range, the uncoupled system is assumed to possess a normally hyperbolic invariant manifold diffeomorphic to a three-sphere, which bounds a strictly convex domain, and whose stable and unstable invariant manifolds coincide. The Hamiltonian flow on the three-sphere is equivalent to the Reeb flow for the induced contact form. The strict convexity condition implies that the contact structure on the three-sphere is tight. When a small, generic coupling is added to the system, the normally hyperbolic invariant manifold is preserved as a three-sphere, and the stable and unstable manifolds split, yielding transverse intersections. We show that there exist trajectories that follow any prescribed collection of invariant tori and Aubry–Mather sets within some global section of the flow restricted to the three-sphere. In this sense, we say that the perturbed system exhibits global diffusion on the tight three-sphere.

A Necessary Condition of Nonminimal Aubry–Mather Sets for Monotone Recurrence Relations

In this paper, we show that a necessary condition for nonminimal Aubry–Mather sets of monotone recurrence relations is that the set of all Birkhoff minimizers with some irrational rotation number does not constitute a foliation, i.e. the gaps of the minimal Aubry–Mather set are not filled up with Birkhoff minimizers.

Mather sets for sequences of matrices and applications to the study of joint spectral radii

The joint spectral radius of a compact set of d-times-d matrices is defined ?to be the maximum possible exponential growth rate of products of matrices drawn from that set. In this article we investigate the ergodic-theoretic structure of those sequences of matrices drawn from a given set whose products grow at the maximum possible rate. This leads to a notion of Mather set for matrix sequences which is analogous to the Mather set in Lagrangian dynamics. We prove a structure theorem establishing the general properties of these Mather sets and describing the extent to which they characterise matrix sequences of maximum growth. We give applications of this theorem to the study of joint spectral radii and to the stability theory of discrete linear inclusions. These results rest on some general theorems on the structure of orbits of maximum growth for subadditive observations of dynamical systems, including an extension of the semi-uniform subadditive ergodic theorem of Schreiber, Sturman and Stark, and an extension of a noted lemma of Y. Peres. These theorems are presented in the appendix.

Diffusion along transition chains of invariant tori and Aubry–Mather sets

AbstractWe describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus; (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map; (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains (i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence); (iv) the transition chains of tori are interspersed with gaps created by resonances; (v) within each gap there is prescribed a finite collection of Aubry–Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry–Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.

Difference approximation to Aubry–Mather sets of the forced Burgers equation

Jauslin–Kreiss–Moser and W E made clear the connection between the Aubry–Mather theory and the inviscid forced Burgers equation with a -periodic forcing term and established the smooth approximation of -periodic entropy solutions of the PDE. This paper presents results of a difference approximation to the Aubry–Mather sets. We prove the convergence of the Lax–Friedrichs scheme for the -periodic entropy solutions. This result leads to difference approximations of the corresponding effective Hamiltonian and -periodic viscosity solutions of the Hamilton–Jacobi equation. We numerically construct the Aubry–Mather sets through the approximate entropy solutions, based on the dynamical properties of the Aubry–Mather sets.

On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.

Ghost circles in lattice Aubry–Mather theory

Monotone lattice recurrence relations such as the Frenkel–Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps. Such recurrence relations often admit a variational structure, so that the solutions x : Z d → R are the stationary points of a formal action function W ( x ) . Given any rotation vector ω ∈ R d , classical Aubry–Mather theory establishes the existence of a large collection of solutions of ∇ W ( x ) = 0 of rotation vector ω. For irrational ω, this is the well-known Aubry–Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the parabolic gradient flow d x d t = − ∇ W ( x ) and we will prove that every Aubry–Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called ‘ghost circle’. The existence of these ghost circles is known in dimension d = 1 , for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry–Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution.

On the integrability of Tonelli Hamiltonians

In this article we discuss a weaker version of Liouville’s Theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced. Moreover, we prove that on the n n -dimensional torus this weaker condition implies classical integrability in the sense of Liouville. The main idea of the proof consists in relating the existence of independent integrals of motion of a Tonelli Hamiltonian to the “size” of its Mather and Aubry sets. As a byproduct we point out the existence of “non-trivial” common invariant sets for all Hamiltonians that Poisson-commute with a Tonelli Hamiltonian.

Aubry sets vs. Mather sets in two degrees of freedom

Let L be an autonomous Tonelli Lagrangian on a closed manifold of dimension two. We prove that for a large set of cohomology classes, the Aubry set consists of periodic orbits.

Boundedness for sublinear reversible systems with a nonlinear damping and periodic forcing term

In this paper, we are concerned with the sublinear reversible systems with a nonlinear damping and periodic forcing term x ″ + f ( x ) g ( x ′ ) + γ | x | α − 1 x = p ( t ) , where f ( x ) , p ( t ) are odd functions, p ( t ) is smooth 1-periodic function and γ ≠ 0 is a constant. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, we show the existence of Aubry–Mather sets as well.