Unfolding the torus: Oscillator geometry from time delays
A method of plotting the trajectories of systems of weakly coupled oscillators is presented. The considered system of \(n\) weakly coupled oscillators has an attracting invariant \(n\)-dimensional torus \(T^ n\) and the attractor as its subset. The flow on \(T^ n\) admits a suitable global section which is an \((n-1)\)-dimensional torus \(T^{n-1}\). Thus the dynamics of \(n\) coupled oscillators is reduced to the study of the Poincaré map \(P: T^{n-1}\to T^{n-1}\), which may have a very complicated, e.g. chaotic, behavior of orbits. Even for three oscillators, a standard projection of \(T^ 2\) onto \(\mathbb{R}^ 2\) is not injective. This makes it difficult to observe the structure of the attractor. An algorithm suggested by the authors allows a direct measurement of angles on \(T^ 2\) providing a possibility of 1-to-1 projection of \(P\) onto the ``unfolded torus''. In the case where the attractor is of knot type, its knot type is easily seen in the projection. Numerical and electronic implementations of the suggested algorithm along with a numerical study of the attractor are given.
numerical and electronic implementations
scientific article; zbMATH DE number 490597