Chaotic Motion in the Breathing Circle Billiard

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.

Aubry-Mather theory for contact Hamiltonian systems II

<p style='text-indent:20px;'>In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems <inline-formula><tex-math id="M1">\begin{document}$ H(x,u,p) $\end{document}</tex-math></inline-formula> with certain dependence on the contact variable <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula>. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set <inline-formula><tex-math id="M3">\begin{document}$ \tilde{\mathcal{S}}_s $\end{document}</tex-math></inline-formula> consists of <i>strongly</i> static orbits, which coincides with the Aubry set <inline-formula><tex-math id="M4">\begin{document}$ \tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show <inline-formula><tex-math id="M5">\begin{document}$ \tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}} $\end{document}</tex-math></inline-formula> in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of <inline-formula><tex-math id="M6">\begin{document}$ H $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M7">\begin{document}$ u $\end{document}</tex-math></inline-formula> fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the <i>minimal</i> viscosity solution and <i>non-minimal</i> ones.</p>

Optimal results in Lorentzian Aubry–Mather theory

This article complements the Lorentzian Aubry–Mather Theory in Suhr (Geom Dedicata 160:91–117, 2012; J Fixed Point Theory Appl 21:71, 2019) by giving optimal multiplicity results for the number of maximal invariant measures. As an application the optimal Lipschitz continuity of the time separation on the Abelian cover is established.

Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom

Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. This book provides the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom). This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. The book follows Mather's strategy but emphasizes a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, the book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.

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.

A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level

In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $E_{L}^{-1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale property in $E_{L}^{-1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_{L}^{-1}(c)$). The proof requires the use of well-known results from Aubry – Mather theory.