INTRODUCTION: NONLINEAR OSCILLATORS

In this paper we describe recent work on the classification of knotted periodic orbits in periodically forced nonlinear oscillators, specifically Duff ing's equation and the pendulum (Josephson junction) equation. After reviewing the Birman & Williams template construction, elementary knot theory and kneading theory for one dimensional maps, we give existence and uniqueness results for families of torus knots which arise in these two oscillators. We indicate how these results enable one to study bifurcation sequences occurring during the creation of complicated invariant sets such as Smale's horseshoe, and how they imply infinitely many distinct “routes to chaos” even in families of two dimensional maps such as that of Hénon. We also include a number of general results on knot and link types, including iterated knots, cablings and prime knots.

Our aim is two-fold; to introduce new and useful techniques to the field of “applied” dynamical systems, and to suggest new applications and sources to the “pure” topologist or knot theorist. In keeping with its introductory nature, the paper is informal and many technical details are omitted; we hope to convey the spirit of our methods and provide a sampling of the results they have yielded so far. In this respect the present paper can be viewed as a guided tour of a series of recent papers which contain full proofs of the results sketched here as well as more complete technical and background information (Birman & Williams [1983a,b], Williams [1983], Franks & Williams [1985a,b], Holmes & Williams [1985], Holmes [1986a,b]).