Knotted periodic orbits in dynamical systems—I: Lorenz's equation

THIS PAPER is the first in a series which will study the following problem. We investigate a system of ordinary differential equations which determines a flow on the 3-sphere S3 (or R3 or ultimately on other 3-manifolds), and which has one or perhaps many periodic orbits. We ask: can these orbits be knotted? .What types of knots can occur? What are the implications? Knotted periodic orbits in dynamical systems do not appear to have been systematically studied, although there is one very well known example. Let (x,, x2, x3, x4) be rectangular coordinates in R4 and let S3 C R4 be the subset of points satisfying Ej=, c? = 1. Let (p, 4) be a pair of coprime integers, and consider the system of ordinary differential equations:

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