Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

TBA

Let, for every positive integer d, a tuple of events A_1,...,A_d be given. Let X_d be the number of events that occur. We state new sufficient conditions for the following extremal independence property: |P(X_d=0)-\prod_{i=1}^d(1-P(A_i))|\to 0. These conditions imply a series of results on asymptotic distributions of certain maximum statistics. In particular, for the maximum number X_n of cliques sharing one vertex in G(n,p), we find sequences a_n and b_n such that (X_n-a_n)/b_n converges in distribution to a standard Gumbel random variable.

This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a ``hole" of positive measure in the phase space which some portion of points in phase space escapes through that hole at each iterate of the dynamical system map. Here, we study the escape rate (a quantity that presents at what rate points in phase space escape through the hole) and various estimations of the escape rate of an open dynamical system. We uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Moreover, exact computations of escape rate and its estimations are present for the skewed tent map and Arnold’s cat map.

In the second part of the thesis, we study physical billiards where the moving particle has a finite nonzero size. In contrast to mathematical billiards where a trajectory is excluded when it hits a corner point of the boundary, in physical billiards reflection of the physical particle (a ball) off a visible corner point is well-defined. Initially, we study properties of such reflections in a physical billiards. Our results confirm that the reflection considered in the literature about physical billiards are indeed no-slip friction-free (elastic) collisions.

In the third part of the thesis, we study physical Ehrenfests' wind-tree models, where we show that physical wind-tree models are dynamically richer than the well-known Lorentz gas model. More precisely, when we replace the point particle by a physical one (a ball), the wind-tree models show a new superdiffusive regimes that never been observed in any other model such as Lorentz gas.

Finally, we prove that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle.

This thesis focuses on analyzing the physics and designing multiscale methods for nonlinear dynamics in mechanical systems, such as those in astronomy. The planetary systems (e.g. the Solar System) are of great interest as rich dynamics of different scales contribute to many interesting physics. Outside the Solar System, a bursting number of exoplanets have been discovered in recent years, raising interest in understanding the effects of the spin dynamics to the habitability. In part I of this thesis, we investigate the spin dynamics of circumbinary exoplanets, which are planets that orbit around stellar binaries. We found that habitable zone planets around the stellar binaries in near coplanar orbits may hold higher potential for stable climate compared to their single star analogues. And in terms of methodology, secular theory of the slow dominating dynamics is calculated via averaging. Beyond analyzing the dynamics mathematically, to simulate the spin-orbit dynamics for long term accurately, symplectic Lie-group (multiscale) integrators are designed to simulate systems consisting of gravitationally interacting rigid bodies in part II of the thesis. Schematically, slow and fast scales are tailored to compose efficient algorithms. And the integrators are tested via our package GRIT. For the systems that are almost impossible to simulate (e.g. the Solar System with the asteroid belt), how can we understand the dynamics from the observations? In part III, we consider the learning and prediction of nonlinear time series purely from observations of symplectic maps. We represent the symplectic map by a generating function, which we approximate by a neural network (hence the name GFNN). And we will prove, under reasonable assumptions, the global prediction error grows at most linearly with long prediction time as the prediction map is symplectic.