Rigorously Verified Numerics for High Dimensional Dynamics

点击翻译按钮获取中文摘要

Nonlinear dynamics is ubiquitous in the natural world. It is pervasive in biology, from the electrophysiological properties of neurons, via the spiralling waves in contracting heart muscles, to gene regulatory networks. It is pervasive in physics, from the swirling motions in fluid flows, via the creation of complex patterns in materials, to the harmonious motions of celestial bodies. It is pervasive in chemistry, from the rich reaction kinetics phenomena, via the chemical basis of morphogenesis at the origin of patterns on animals, to the complicated biochemistry in the living cell. Mathematically, these beautiful phenomena are described by nonlinear dynamical systems in the form of ordinary differential equations (ODEs), partial differential equations (PDEs) and delay differential equations (DDEs). While the presence of nonlinearities complicates the analysis, the challenges are even greater for PDEs and DDEs, which are naturally defined on infinite dimensional function spaces. Thanks to the availability of high-performance computers and the rapid development of sophisticated software, numerical simulations have become the primary tool used by scientists to study these models. However, the prevalence of computations in modern day science naturally leads to the fundamental question of validity of the outputs. Even for ODEs, this question arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical simulations) yield widely diverging outcomes. To address this issue, the recent field of rigorous numerics emerged at the intersection of pure and applied mathematics. Rigorous numerics draws inspiration from scientific computing, nonlinear analysis, numerical analysis, applied topology, functional analysis and approximation theory. In this research proposal, we propose to introduce novel and exciting techniques within the field of rigorous numerics to study cutting-edge problems in finite and infinite dimensional dynamical systems, with an emphasis on problems arising in fluids, pattern formation and celestial mechanics. More precisely, we are interested in answering each of the following questions. Can we rigorously control the errors made when computing solutions of Cauchy problems of PDEs? Can we develop rigorous computations for connecting orbits in infinite dimensions, which are crucial for our understanding of pattern formation? Can we mathematically demonstrate the existence of chaos in infinite dimensional continuous dynamical systems? Can we prove a long standing conjecture concerning the continuation of choreographies in celestial mechanics? In the long term, can we show that Navier-Stokes do not develop singularities as time evolves for a certain class of initial data, or can we identify (numerically-perhaps with rigorous validation) an initial data for which it does?