We describe topological methods for the efficient, rigorous computation of dynamical systems. In particular, we indicate how Conley's Fundamental Decomposition Theorem is naturally related to combinatorial approximations of dynamical systems. Furthermore, we show that computations of Morse decompositions and isolating blocks can be performed efficiently. We conclude with examples indicating how these ideas can be applied to finite- and infinite-dimensional discrete and continuous dynamical systems.

Introduction

This paper is an expository article on using topological methods for the efficient, rigorous computation of dynamical systems. Of course, since its inception the computer has been used for the purpose of simulating nonlinear models. However, in recent years there has been a rapid development in numerical methods specifically designed to study these models from a dynamical systems point of view, that is, with a particular emphasis on the structures which capture the long-term or asymptotic states of the system. At the risk of greatly simplifying these results, this work has followed two themes: indirect methods and direct methods.

The indirect methods are most closely associated with simulations, and as such are extremely important because they tend to be the cheapest computationally. The emphasis is on developing numerical schemes whose solutions exhibit the same dynamics as the original system: for example, if one is given a Hamiltonian system, then it is reasonable to want a numerical method that preserves the integrals of the original system.

A key aspect of the power of the Schwarz–Christoffel transformation (indeed, a large part of the motivation for this book) is its remarkable flexibility in adapting to a wide variety of situations, not all of which superficially seem to involve conformal maps or even polygons. The essence of SC mapping is to treat the corners exactly; if the rest of the problem is simple, nothing else is needed. What emerges from applications of this principle is that Schwarz–Christoffel mapping is not just a mapping technique but a distinctive way of thinking about problems of potential theory in the plane.

Let us reconsider the fundamental SC philosophy for constructing a map f (z). For the half-plane, we required f′ to have piecewise constant argument along the boundary because then the image under f has straight lines with corners. For other canonical domains, we need to modify this requirement slightly. For example, as we follow the boundary of the unit disk, a constant argument for f′ does not lead to a straight-line image. However, if g(z) is a function that “straightens out” the original domain boundary, then f′/g′ will have the appropriate piecewise-constant argument. This fact is especially convenient because we can also use powers of g(z) – g(zk) to create wedges that have the right jumps.

Conformal mapping in general, and Schwarz–Christoffel mapping in particular, are fascinating and beautiful subjects in their own rights. Nevertheless, the history of conformal mapping is driven largely by applications, so it is appropriate to consider when and how SC mapping can be used in practical problems.

It is not our intent in this final chapter to recount every instance in which Schwarz–Christoffel mapping has been brought to bear. Rather, after a brief look at a few areas full of such examples, we describe some situations in which SC ideas can be applied in ways that are computational and perhaps not transparent. The most famous application of conformal mapping is to Laplace's equation, and we devote three sections to it. Beyond this it is clear that Schwarz–Christoffel mapping has a small but important niche in applied mathematics and science.

In applications it is common to pose a physical problem in the z-plane, which maps to a canonical region in the w-plane. This convention runs counter to our discussion in the earlier chapters, in which w was the plane of the polygon. In the following sections we attempt to be consistent with established applications literature where appropriate.

Why use Schwarz–Christoffel maps?

Schwarz–Christoffel mapping is an incomparably effective tool for a very specific sort of problem. The most natural and satisfying application is the solution of Laplace's equation in the plane with piecewise constant (and in the case of derivative conditions, homogeneous) boundary conditions.

Polygons

For the rest of this book, a (generalized) polygon Γ is defined by a collection of vertices w1, …, wn and real interior angles α1π, …, αnπ. It is convenient for indexing purposes to define wn+1 = w1 and w0 = wn. The vertices, which lie in the extended complex plane C ∪ {∞}, are given in counterclockwise order with respect to the interior of the polygon (i. e., locally the polygon is “to the left” as one traverses the side from wk to wk+1).

The interior angle at vertex k is defined as the angle swept from the outgoing side at wk to the incoming side. If |wk| < ∞, we have αk ∈ (0, 2]. If αk = 2, the sides incident on wk are collinear, and wk is the tip of a slit. The definition of the interior angle is applied on the Riemann sphere if wk = ∞. In this case, αk ∈ [–2, 0]. See Figure 2.1. Specifying αk is redundant if wk and its neighbors are finite, but otherwise αk is needed to determine the polygon uniquely.

In addition to the preceding restrictions on the angles αk, we require that the polygon make a total turn of 2π.

In the autumn of 1978, Peter Henrici took leave from the ETH in Zurich to visit the Numerical Analysis Group at Stanford University. The second author, then a graduate student, asked Henrici if he might propose a project in the area of computational complex analysis. Henrici's suggestion was, why don't you see what you can do with the Schwarz–Christoffel transformation?

For months thereafter LNT spent all of every weekend working on SC mapping at the computer terminals of the Stanford Linear Accelerator Center. This brief but intense project led to one of the first technical reports ever printed in TEX, which was published in the first issue of the SIAM Journal on Scientific and Statistical Computing; to the FORTRAN package SCPACK; and to a lasting love of numerical conformal mapping. In the following years it led further to extensions and applications of SC ideas carried out in collaboration with various people, including Alan Elcrat and Frédéric Dias on free-streamline flows, Ruth Williams on oblique derivative problems, and Louis Howell on modified formulas for elongated regions.

By the early 1990s, LNT was a faculty member at Cornell University and the first author was a graduate student. We worked together on a number of topics from hydrodynamic stability to “Can one hear the shape of a drum?” but the subject we kept coming back to was Schwarz–Christoffel mapping. Once again it started with a brief suggestion.