Aubry-Mather Theory Research Articles

Aubry–Mather theory on graphs

We formulate Aubry–Mather theory for Hamiltonians/Lagrangians defined on graphs, study the structure of minimizing measures, and discuss the relationship with weak KAM theory developed in Siconolfi and Sorrentino (2018 Anal. PDE 1 171–211). Moreover, we describe how to transport and interpret these results on networks.

Quantum tomographic Aubry–Mather theory

In this paper, we study the quantum analog of the Aubry–Mather theory from a tomographic point of view. In order to have a well-defined real distribution function for the quantum phase space, which can be a solution for variational action minimizing problems, we reconstruct quantum Mather measures by means of inverse Radon transform and prove that the resulting tomograms, which are fair and non-negative distribution functions, are also solutions of the quantum Mather problem and, in the semi-classical sense, converge to the classical Mather measures.

Discrete approximation of stationary Mean Field Games

<p style='text-indent:20px;'>In this paper, we focus on stationary (ergodic) mean-field games (MFGs). These games arise in the study of the long-time behavior of finite-horizon MFGs. Motivated by a prior scheme for Hamilton–Jacobi equations introduced in Aubry–Mather's theory, we introduce a discrete approximation to stationary MFGs. Relying on Kakutani's fixed-point theorem, we prove the existence and uniqueness (up to additive constant) of solutions to the discrete problem. Moreover, we show that the solutions to the discrete problem converge, uniformly in the nonlocal case and weakly in the local case, to the classical solutions of the stationary problem.</p>

Depinning transition of travelling waves for particle chains

In this paper we apply Aubry–Mather theory for equilibria of 1D Hamiltonian lattice systems and the theory of invariant ordered circles to investigate the depinning transition of travelling waves for particle chains. Assume A < B are two critical values such that the particle chain has three homogeneous equilibria if the driving force . It is already known that there exist transition thresholds of the driving force such that the particle chain has stationary fronts but no travelling fronts for and travelling fronts but no stationary fronts if or . The novelty of our approach is that we prove the transition threshold () coincides with the upper (lower) limit of the upper (lower) depinning force as the rotation number tends to zero from the right. Based on this conclusion, we demonstrate that when the driving force , besides stationary fronts there are various kinds of equilibria with rotation numbers close to zero such that the spatial shift map has positive topological entropy on the set of equilibria. Furthermore, we give a necessary and sufficient condition for the absence of propagation failure, i.e. , in terms of a minimal foliation. Finally we show that are continuous with respect to potential functions in C 1 topology.

Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum

We investigate a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation. Based on KAM theory, we prove that there are three regions (corresponding to different energies) occupied by quasi-periodic solutions in phase space when the periodic excitation is small. Moreover, the rotational quasi-periodic motion is maintained when the perturbation gets larger. The existence of subharmonic periodic solutions is obtained by the Aubry–Mather theory and the boundedness of all solutions is followed by the fact that there exist abundant invariant tori near infinity. To study the homoclinic bifurcation of this system, we present a numerical method to compute the discontinuous invariant manifolds accurately, which provides a useful tool for the study of invariant manifolds under the effect of impacts.

High degeneracy of effective Hamiltonian in two dimensions

Consider the effective Hamiltonian \overline{H}(p) associated with the mechanical Hamiltonian H(p,x)=\frac12|p|^2+V(x) . We prove that for generic V , \overline{H} is piecewise one-dimensional in a dense open set in two dimensions using Aubry–Mather theory.

On the global dynamical properties of a Fermi–Ulam model

We consider a Fermi–Ulam model describing a particle which moves freely between two parallel walls and is reflected elastically when hitting the walls. The walls are supposed to oscillate according to a given regular periodic function. We show that in high energy situation, the global dynamics of this model can be described by an exact area-preserving monotone twist map. Then the existence of abundant periodic orbits and quasi-periodic orbits is proved based on the Aubry–Mather theory.

KAM and Geodesic Dynamics of Blackholes

In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of bound geodesic dynamics of a perturbed blackhole background. The general theories apply mainly to two observable phenomena: the photon shell (unstable bound spherical orbits) and the quasi-periodic oscillations. We discover there is a gap structure in the photon shell that can be used to reveal information of the perturbation.

The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures

Effective Hamiltonians arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry–Mather theory. In Aubry–Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

Twist dynamics and Aubry-Mather sets around a periodically perturbed point-vortex

We consider the model of a point-vortex under a periodic perturbation and give sufficient conditions for the existence of generalized quasi-periodic solutions with rotation number. The proof relies on Aubry-Mather theory to obtain the existence of a family of minimal orbits of the Poincaré map associated to the system.

The selection problem for some first-order stationary Mean-field games

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

MATHER measures for space–time periodic nonconvex Hamiltonians

In Gomes (Nonlinearity 15(3):581–603, 2002) developed techniques and tools with the purpose of extending the Aubry–Mather theory in a stochastic setting, namely he proved the existence of stochastic Mather measures and their properties. These results were also extended in the time-dependent setting in the doctoral thesis of the Guerra-Velasco (http://132.248.9.195/ptd2015/abril/506015252/Index.html, 2015). However, to construct analogs to the Aubry–Mather measures for nonconvex Hamiltonians, it is necessary to use the adjoint method introduced by Evans (Arch Ration Mech Anal 197:1053–1088, 2010) and Tran (Calc Var Partial Differ Equ 41:301–319, 2011); the construction of the measures is in Cagnetti et al. (SIAM J Math Anal 43(6):2601–262, citelink2011CGT). The main goal of this paper is to construct Mather measures for space–time periodical nonconvex Hamiltonians using the techniques in [10, 21] and [7]. Moreover, we also will prove that there is only one value, such that the viscous Hamilton–Jacobi equation has a smooth periodic solution unique up to an additive constant.

Aubry–Mather theory for Lorentzian manifolds

We introduce a version of Aubry–Mather theory for the length functional of causal curves in compact Lorentzian manifolds. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions of the Mather’s graph theorem. A class of examples, the Lorentzian Hedlund examples, shows the optimality of the obtained results.

Rate of Convergence in Periodic Homogenization of Hamilton–Jacobi Equations: The Convex Setting

We study the rate of convergence of $u^\epsilon$, as $\epsilon \to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^\epsilon$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm (C)_\epsilon} \qquad \begin{cases} u_t^\epsilon+H\left(\frac{x}{\epsilon},Du^\epsilon\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^\epsilon(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} and \begin{equation*} {\rm (C)} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} respectively. We assume that the Hamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is $\mathbb{Z}^n$-periodic in the $y$ variable, and the initial data $g$ is bounded and Lipschitz continuous.

Aubry–Mather Theory for Contact Hamiltonian Systems

In this paper, we focus on action-minimizing methods for contact Hamiltonian systems. Based on implicit variational principles introduced in Wang et al. (Nonlinearity 30:492–515, 2017), we generalize some fundamental results of Aubry–Mather theory and weak KAM theory from Hamiltonian systems to contact Hamiltonian systems with moderate increasing on the contact variable.

Applications of variable discounting dynamic programming to iterated function systems and related problems

We study existence and uniqueness of the fixed point solutions of a large class of nonlinear variable discounted transfer operators associated to a sequential decision-making process. We establish regularity properties of these solutions, with respect to the immediate return and the variable discount. In addition, we apply our methods to reformulating and solving, in the setting of dynamic programming, some central variational problems on the theory of iterated function systems, Markov decision processes, discrete Aubry–Mather theory, Sinai–Ruelle–Bowen measures, fat solenoidal attractors, and ergodic optimization.

Mather theory and symplectic rigidity

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when $H$ fails to be convex, even for nearly integrable $H$. Parts of these results extend work by Viterbo and Vichery. Using ideas due to Entov-Polterovich we also detect interesting invariant measures for $\phi_H$ by studying a generalization of the symplectic shape of sublevel sets of $H$. This approach differs from the first one in that it works also for $(M,\omega)$ in which every compact subset can be displaced. We present applications to Hamiltonian systems on $\mathbb{R}^{2n}$ and twisted cotangent bundles.

Continuity of depinning force

Monotone recurrence relations give rise to a class of dynamical systems on the high-dimensional cylinder which generalizes the class of monotone twist maps. A solution of a monotone recurrence relation corresponds to an equilibrium of the generalized Frenkel–Kontorova (FK) model. The Aubry–Mather theory for monotone recurrence relations says that for each ω∈R there is a Birkhoff minimizer with rotation number ω. Whether all Birkhoff minimizers of rotation number ω form a foliation is a question like whether there is an invariant circle with rotation number ω for a monotone twist map. In this paper we give a criterion for the existence of minimal foliations and study its continuity.The depinning force Fd(ω), depending on rotation numbers, is a critical value of the driving force for the FK model, under which there are Birkhoff equilibria and hence the system is pinned, and above which there are no Birkhoff equilibria of rotation number ω and the system is sliding. We show that all Birkhoff minimizers with irrational rotation number ω form a foliation if and only if Fd(ω)=0. If ω=p/q is rational, then Fd(p/q)=0 if and only if all (p,q)-periodic Birkhoff minimizers constitute a foliation. Moreover, we show that Fd(ω) is continuous at irrational ω and Hölder continuous at Diophantine points, and it also depends continuously on system parameters.

Nonsmooth convex caustics for Birkhoff billiards

This paper is devoted to the examination of the properties of the string construction for the Birkhoff billiard. Based on purely geometric considerations, string construction is suited to provide a table for the Birkhoff billiard, having the prescribed caustic. Exploiting this framework together with the properties of convex caustics, we give a geometric proof of a result by Innami first proved in 2002 by means of Aubry-Mather theory. In the second part of the paper we show that applying the string construction one can find a new collection of examples of $C^2$-smooth convex billiard tables with a non-smooth convex caustic.

Denjoy minimal sets and Birkhoff periodic orbits for non-exact monotone twist maps

A non-exact monotone twist map φ¯F is a composition of an exact monotone twist map φ¯ with a generating function H and a vertical translation VF with VF((x,y))=(x,y−F). We show in this paper that for each ω∈R, there exists a critical value Fd(ω)≥0 depending on H and ω such that for 0≤F≤Fd(ω), the non-exact twist map φ¯F has an invariant Denjoy minimal set with irrational rotation number ω lying on a Lipschitz graph, or Birkhoff (p,q)-periodic orbits for rational ω=p/q. Like the Aubry–Mather theory, we also construct heteroclinic orbits connecting Birkhoff periodic orbits, and show that quasi-periodic orbits in these Denjoy minimal sets can be approximated by periodic orbits. In particular, we demonstrate that at the critical value F=Fd(ω), the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves.