Inversely Unstable Solutions of Two-Dimensional Systems on Genus-p Surfaces and the Topology of Knotted Attractors

In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by studying the appearance of inversely unstable solutions within each invariant set.Comment: 19 pages with 20 figures, AMS La-TeX, to be published in International Journal of Bifurcation and Chao

Global analysis of dynamical systems on low-dimensional manifolds.

The interaction of topology and dynamics has attracted a great deal of attention from numerous mathematicians. This thesis is devoted to the study of dynamical systems on low-dimensional manifolds. In the order of dimensions, we first look at the case of two-manifolds (surfaces) and derive explicit differential equations for dynamical systems defined on generic surfaces by applying elliptic and automorphic function theory to uniformise the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series, we will determine general meromorphic systems on a fundamental domain in the upper half plane, the solution trajectories of which 'roll up' onto an appropriate surface of any given genus. Meanwhile, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by analyzing the topology of inversely unstable solutions contained within each invariant set. Then the thesis concerns a study of three-dimensional systems. We give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tubular neighbourhood of the braid. The second one, generating chaotic systems, is realized by modelling the Smale horseshoe. Moreover, we shall consider the analytical and topological structure of systems on 2- and 3- manifolds. By considering surgery operations, such as Dehn surgery, Heegaard splittings and connected sums, we shall show that it is possible to obtain systems with 'arbitrarily strange' behaviour, Le., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We will also consider diffeomorphisms which are defined on closed 3-manifolds and contain generalized Smale solenoids as the non-wandering sets. Motivated by the result in [Jiang, Ni and Wang, 2004], we will investigate the possibility of generating dynamical systems containing an arbitrary number of solenoids on any closed, orientable 3-manifold. This shall also include the study of branched coverings and Reeb foliations. Based on the intense development from four-manifold theory recently, we shall consider four-dimensional dynamical systems at the end. However, this part of the thesis will be mainly speculative

Statistical Equilibrium of Circulating Fluids

We are investigating the inviscid limit of the Navier-Stokes equation, and we find previously unknown anomalous terms in Hamiltonian, Dissipation, and Helicity, which survive this limit and define the turbulent statistics. We find various topologically nontrivial configurations of the confined Clebsch field responsible for vortex sheets and lines. In particular, a stable vortex sheet family is discovered, but its anomalous dissipation vanishes as $\sqrt{\nu}$. Topologically stable stationary singular flows, which we call Kelvinons, are introduced. They have a conserved velocity circulation $\Gamma_\alpha$ around the loop $C$ and another one $\Gamma_\beta$ for an infinitesimal closed loop $\tilde C$ encircling $C$, leading to a finite helicity. The anomalous dissipation has a finite limit, which we computed analytically. The Kelvinon is responsible for asymptotic PDF tails of velocity circulation, \textbf{perfectly matching numerical simulations}. The loop equation for circulation PDF as functional of the loop shape is derived and studied. This equation is \textbf{exactly} equivalent to the Schr\"odinger equation in loop space, with viscosity $\nu$ playing the role of Planck's constant. Kelvinons are fixed points of the loop equation at WKB limit $\nu \rightarrow 0$. The anomalous Hamiltonian for the Kelvinons contains a large parameter $\log \frac{|\Gamma_\beta|}{\nu}$. The leading powers of this parameter can be summed up, leading to familiar asymptotic freedom, like in QCD. In particular, the so-called multifractal scaling laws are, as in QCD, modified by the powers of the logarithm.Comment: 246 pages, 96 figures, and six appendixes. Submitted to Physics Reports. Revised the energy balance analysis and discovered asymptotic freedom leading to powers of logarithm of scale modifying K41 scaling law

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Active Materials

What is an active material? This book aims to redefine perceptions of the materials that respond to their environment. Through the theory of the structure and functionality of materials found in nature a scientific approach to active materials is first identified. Further interviews with experts from the natural sciences and humanities then seeks to question and redefine this view of materials to create a new definition of active materials

Active Materials

What is an active material? This book aims to redefine perceptions of the materials that respond to their environment. Through the theory of the structure and functionality of materials found in nature a scientific approach to active materials is first identified. Further interviews with experts from the natural sciences and humanities then seeks to question and redefine this view of materials to create a new definition of active materials