Introduction to Rigorous Numerics in Dynamics: General functional analytic setup and an example that forces chaos

In this paper some basic concepts of rigorous computing in a dynamical systems context are outlined. We often simulate dynamics on a computer, or calculate a numerical solution to a partial differential equation. This gives very detailed, stimulating information. However, mathematical insight and impact would be much improved if we can be sure that what we see on the screen genuinely represents a solution of the problem. In particular, rigorous validation of the computations allows such objects to be used as ingredients of theorems. The past few decades have seen enormous advances in the development of computer-assisted proofs in dynamics. One approach is based on a functional analytic setup. The goal of this paper is to introduce the ideas underlying this rigorous computational method. As the central example we use the problem of finding a particular periodic orbit in a nonlinear ordinary differential equation that describes pattern formation in fluid dynamics. This simple setting keeps technicalities to a minimum. Nevertheless, the rigorous computation of this single periodic orbit implies chaotic behavior via topological arguments (in a sense very similar to “period 3 implies chaos” for interval maps).