Nonlinear oscillators, iterated maps, symbolic dynamics, and knotted orbits

We illustrate some recent developments in the theory of dynamical systems. Concentrating on periodically forced second-order ordinary differential equations, and using the Josephson (pendulum) and Duffing equations as examples, we show how the method of symbolic dynamics allows one to study complicated (chaotic) invariant sets in the Poincaré maps of such systems. Under certain conditions, these two-dimensional invertible maps can be reduced to one-dimensional (noninvertible) maps on the interval or circle. In this situation, a fairly complete kneading theory is available and one can use it to study bifurcation sequences occurring as parameters vary. We apply these ideas to the Josephson equation. In contrast to reducing the flow to a lower dimensional Poincaré map, Birman and Williams have exploited reduction to a semiflow on a branched two-dimensional manifold by collapsing orbits along stable contracting directions. Using their fundamental result that the knot types of periodic orbits (closed curves in 3-space) are preserved under this collapse--we study some of the knots arising in the Duffing equation. Since the knot type of a periodic orbit is invariant under continuous deformations, one can use it to characterize families of such orbits in parameterized equations. This notion permits us to follow bifurcating sequences of orbits for Poincaré maps of heavily damped (almost one-dimensional) and lightly damped (almost area-preserving) systems and to show that certain "universal," one-dimensional bifurcation sequences are completely reversed for area-preserving maps.