Unfolding the torus: Oscillator geometry from time delays

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 of n weakly coupled oscillators there is an attracting invariant n-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 of n coupled oscillators can thus be reduced, in principle, to the study of Poincaré maps of the (n-1)-dimensional torus. This paper gives a practical algorithm for measuring the n-1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate on n=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.