DUT Dynamics Seminar

We consider the breathing circle billiard, namely the free motion of a point particle inside a disk whose radius varies periodically in time, with elastic reflections at the moving boundary. Since angular momentum is preserved, fixing a value $c$ reduces the dynamics to a two-dimensional exact symplectic map on a cylinder.

In the high-energy regime, the corresponding map is generated by a diagonally periodic twist generating function $h_c$. We study the small angular momentum regime as a perturbation of the limiting case $c = 0$, which corresponds to the Fermi-Ulam dynamics along a diameter. Using this perturbative structure and a quantitative version of Mather converse-KAM criterion, we exclude invariant Lipschitz graphs for suitable rotation numbers. Combined with Aubry-Mather theory and Forni’s theorem, this yields positive topological entropy for sufficiently small $c\geq 0$. Our result gives an essential improvement of a previous similar results obtained via the standard Mather converse-KAM criterion.

The Smale horseshoe provides one of the clearest geometric mechanisms by which deterministic systems generate chaotic dynamics. Beginning with a simple process of stretching, contracting and folding a region of the phase space, the horseshoe map produces an invariant Cantor set on which the dynamics is conjugate to the full shift of two symbols. This symbolic description reveals the essential features of chaos: sensitive dependence on initial conditions, dense periodic orbits, and topological transitivity. In the context of Hamiltonian systems, the horseshoe is especially important because it models the dynamics created near transverse intersections of stable and unstable manifolds of hyperbolic saddles. Such intersections generate homoclinic tangles, which are the geometric source of chaotic motion in many conservative systems. Thus, the horseshoe map serves as a bridge between local hyperbolic manifold geometry and chaos. This talk introduces the horseshoe construction as a foundational example before studying how similar structures arise in Hamiltonian systems through homoclinic intersections and separatrix splitting.

In this talk, we will present a stochastically perturbed Kepler problem, which is based on a result of A. Saha (https://link.springer.com/article/10.1007/s10569-025-10265-z). We will first introduce the stochastic Hamilton equations. Next, we consider a stochastic Kepler problem perturbed by a Hamiltonian noise affecting the angular momentum vector. We show that the angular momentum and the Laplace–Runge–Lenz vectors are conserved in magnitude and as a consequence, the distance and speed of the particle follow deterministic dynamics. If time permits, we will also show the Moser regularization, whereby orbits for a fixed negative energy level are transformed to the geodesic flow on the 3-sphere.

Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. Under suitable conditions, the fast variable can be “averaged out” to produce an averaged system, which is easier for analysis and governs the evolution over a long time scale. In this talk, we consider the averaging principle for SDEs.

This talk will give a brief introduction to the regularization theory, with a particular focus on the problems from celestial mechanics, such as the Kepler problem and the restricted three-body problem. Starting with the Kepler problem, we will demonstrate how to manipulate its equations of motion to regularize the singularity. Next, we will show the regularization method from the Hamiltonian perspective. After that, the planar circular restricted three-body problem and its regularization will be introduced. We will understand why this is important for both theoretical and numerical studies. If time permits, the elliptic problem will also be mentioned.

This talk continues our discussion of ergodic theory on horseshoes. We begin with a brief review of the horseshoe map, a classic example in differentiable dynamical systems, and its relationship with symbolic systems. We then recall some basic concepts in ergodic theory, including invariant measures, ergodic measures, and entropy. Finally, we investigate intermediate value properties and multifractal analysis on horseshoes.

This talk begins with a brief overview of KAM theory for Hamiltonian PDEs and the challenges posed by derivative nonlinearities. We then discuss recent progress on full-dimensional invariant tori for a 1D derivative nonlinear wave equation. To control frequency shifts while preserving non-resonance conditions through KAM iteration, we introduce a modified quasi-Töplitz framework unifying Töplitz-Lipschitz and quasi-Töplitz techniques. This yields linearly stable invariant tori with sub-exponentially decaying amplitudes under mild parameter assumptions.

In this talk， we will discuss the existence of full dimensional tori for Hamiltonian PDEs by KAM theory for infinite dimensional Hamiltonian systems.