The topological reconstruction of forced oscillators

Abstract

Periodically forced oscillators are among the simplest dynamical systems capable to display chaos. They can be described by the variables position and velocity, together with the phase of the force. Their phase-space corresponds therefore to R 2 × S 1 . The organization of the periodic orbits can be displayed with braids having only positive crossings. Topological characterization of dynamical systems actually began to be explored in physics on this family of problems. In this work we show that, in general, it is not possible to produce a 3-dimensional imbedding of the solutions of a forced oscillator in terms of differential imbeddings based on sampling the position only. However, it may be possible to uncover a description of the phase variable from the sampled time-series, thus producing a faithful representation of the data. We proceed to formulate new tests in order to check whether proposed imbeddings can be accepted as such. We illustrate the manuscript throughout with an example corresponding to a model of Bénard–Marangoni convection.