Topological Analysis of Chaotic Data
Abstract
Most of the work done in order to describe strange attractors has been metric in nature (computation of dimensions, Lyapunov exponents ...). This approach requires large data sets, and returns little information. In this thesis, we propose a topological characterization of strange attractors. This characterization is based on the determination of the template or knot holder: a branched manifold that supports all periodic orbits embedded in the strange attractor. The template is determined by the topological organization of low period orbits, and permits predictions of higher period orbits. We show how to implement these ideas on experimental time series data. Once the topology of the strange attractor is identified, this fingerprint can be used in model building. The organization of the orbits not only helps to classify dynamical systems: we show how the topological properties of the unstable orbits embedded in the strange attractor determines observational features (e.g. the power spectrum of the time series). Finally, the topological structure of the flow is also used in order to establish selection rules for bifurcations; conditions that remain invariant if the dissipation of the system is changed.
- Publication:
-
Ph.D. Thesis
- Pub Date:
- 1991
- Bibcode:
- 1991PhDT.......221M
- Keywords:
-
- KNOTS;
- Physics: General