Topological Characterization of Reconstructed Attractors Modding Out Symmetries

(Received 24 October 1995, revised 2 May 1996, accepted 8 August 1996)

Abstract
Topological chaxacterization is important in understanding the subtleties of chaotic behaviour. Unfortunately it is based on the knot theory which is only efficiently developed in 3D spaces (namely or in its one-point compactification ). Consequently, to achieve topological characterization, phase portraits must be embedded in 3D spaces, i.e. in a lower dimension than the one prescribed by Takens' theorem. Investigating embedding in low-dimensional spaces is, therefore, particularly meaningful. This paper is devoted to tridimensional systems which are reconstructed in a state space whose dimension is also 3. In particular, an important case is when the system studied exhibits symmetry properties, because topological properties of the attractor reconstructed from a scalar time series may then crucially depend on the variable used. Consequently, special attention is paid to systems with symmetry properties in which specific procedures for topological characterization are developed. In these procedures, all the dynamics are projected onto a so-called fundamental domain, leading us to the introduction of the concept of restricted topological equivalence, i.e. two attractors are topologically equivalent in the restricted sense, if the topological properties of their fundamental domains are the same. In other words, the symmetries are moded out by projecting the whole phase space onto a fundamental domain.