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# Introduction

In September of 1997, Nick Tufillaro discussed some applications of symbolic dynamics to problems in mathematics, physics, and engineering in an IMA Industrial Problems Seminar. This article will discuss some of the applications from Nick's talk.

The mathematical aspects of Nick's discussion included several topics, including kneading theory, horseshoes, and braid analysis. Two systems that he discussed were the quadratic map (and other unimodal maps), and the Rossler system of differential equations. Some problems from physics that Nick discussed were the vibrating string and the Belousov-Zhabotinskii reaction. The goals of an engineering application might include a problem such as detecting bifurcations in noisy experimental data. As an example from engineering, Nick discussed an application of symbolic dynamics in the study of cycle variability in internal combustion engines.

The different emphasis of the applications in the different disciplines is summarized in Table 1. The recent applications in physics and engineering suggest that symbolic dynamics is emerging as a powerful tool in nonlinear system identification.

Table 1: Symbolic dynamics in mathematics, physics, and engineering. The germ of the ideas is the same in each discipline, but the goals and implementations can be very different.

Nick outlined the general strategy for using symbolic dynamics as an analysis tool as follows:

• Identify/specify the topological class/type.
• Set up a ``symbol dynamics'' for that particular class. (The symbolic dynamics are the topological coordinates appropriate for nonlinear problems.)
• Extract as much information as you can based on topology alone before considering metric properties. Identify/measure ``topological parameters'' if possible.

These ideas will be developed below. Another point of Nick's talk was the challenges of collaboration between academia and industry. His comment is worth quoting: ``The different goals and problems can usually not be properly understood or communicated in an afternoon. [It] really takes an immersion in each others' work environment and culture. There is more than a difference in vocabulary stifling effective collaboration.''

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