Rigorous Numerics in Dynamics

Abstract

MotivationNonlinear dynamics shape the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these beautiful phenomena are modelled by nonlinear dynamical systems, mainly in the form of ordinary differential equations (ODEs), partial differential equations (PDEs) and delay differential equations (DDEs). The presence of nonlinearities severely complicates the mathematical analysis of these dynamical systems, and the difficulties are even greater for PDEs and DDEs, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics.Rigorous numerics draws inspiration from the ideas in scientific computing, numerical analysis and approximation theory. In a nutshell, rigorous computations are mathematical theorems formulated in such a way that the assumptions can be rigorously verified on a computer. This requires an a priori setup that allows analysis and numerics to go hand in hand: the choice of function spaces, the choice of the basis functions and Galerkin projections, the analytic estimates, and the computational parameters must all work together to bound the errors due to approximation, rounding and truncation sufficiently tightly for the verification proof to go through. The goal is to provide a mathematically rigorous statement about the validity of a concrete numerical simulation (i.e. not in some asymptotic sense where for example the grid size tends to zero) as interpreted as an approximate solution of the original problem. This complements the field of scientific computing, where the goal is to achieve highly reliable results for very complicated problems. In rigorous computing one is after absolutely reliable results for somewhat less complicated (but still hard) problems.Outside of dynamics, computer-assisted proofs have been used to settle famous open problems. Two prominent examples are the four color theorem [1] and Kepler's densest sphere packing problem [2]. In dynamical systems, an early success is the demonstration of the universality of the Feigenbaum constant [3]. More recently, rigorous numerics were used to prove the existence of the strange attractor in the Lorenz system, which seemed, for decades, tentatively intuitive from computer simulations [4]. This settled the 14th problem in Smale's list of problems for the 21st century (the only other problem from the list that has been solved is the P...

When does a periodic response exist in a periodically forced multi-degree-of-freedom mechanical system?

“…Rigorous numerics (cf. van den Berg and Lessard [46]), estimating the ignored tail of Taylor expansions, is only applicable to specific numerical examples. In applications, therefore, one typically employs an additional numerical method to verify the predictions of perturbation methods.…”

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

“…Our approach is in the functional analytic tradition of Lanford, Eckmann, Wittwer, and Koch whose work on on renormalization theory and the proof of the Feigenbaum conjectures [41,42,43,44] was foundational. For broader surveys of the literature on computer assisted proof in analysis we refer the interested reader to the surveys [45,46,47,48].…”

An analysis of the fourth‐order PDEs – ‘patterns’ and ‘di‐block polymers’

“…Firstly, the existence of standing waves between rolls and hexagonal patterns of the two‐dimensional pattern formation PDE model is given in as the evolutionary equation where u = u ( x , y , t ) and Δ is the two‐dimensional Laplacion. This is a generalization of the Swift–Hohenberg equation given in .…”

“…Depending on the value of the parameter δ which represent a measure of the ratio of the mixture of the polymers and k , the incompatibility of the polymer types, there is a range of stationary states with a three‐dimensional geometry. These have been analyzed and presented using numerical techniques (see reference above). In , for k =0, an energy functional is given by The texts on invariance studies and conservation law methods for PDEs are well known, and some of the relevant ones here are .…”