TOPOLOGICAL CHARACTERIZATION OF RECONSTRUCTED ATTRACTORS MODDING OUT SYMMETRIES

Topological characterization is important in understanding the subtlet ies of chaotic behaviour. Unfortunately it is based on the knot theory which is only efficiently developed in 3D spaces (namely R(3) or in i ts one-point compactification S-3). Consequently, to achieve topologic al characterization, phase portraits must be embedded in 3D spaces, i. e. in a lower dimension than the one prescribed by Takens' theorem. In vestigating embedding in low-dimensional spaces is, therefore, particu larly meaningful. This paper is devoted to tridimensional systems whic h are reconstructed in a state space whose dimension is also 3. In par ticular, an important case is when the system studied exhibits symmetr y properties, because topological properties of the attractor reconstr ucted from a scalar time series may then crucially depend on the varia ble used. Consequently, special attention is paid to systems with symm etry properties in which specific procedures for topological character ization are developed. In these procedures, all the dynamics are proje cted onto a so-called fundamental domain, leading us to the introducti on of the concept of restricted topological equivalence, i.e. two attr actors are topologically equivalent in the restricted sense, if the to pological properties of their fundamental domains are the same. In oth er words, the symmetries are moded out by projecting the whole phase s pace onto a fundamental domain.