Unfolding the torus: Oscillator geometry from time delays

Summary

We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.