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Journal Articles Journal de Physique II Year : 1996

Topological Characterization of Reconstructed Attractors Modding Out Symmetries

G. Gouesbet


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 ${\rm I\!R}^3$ or in its one-point compactification S3). 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.

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Dates and versions

jpa-00248397 , version 1 (01-01-1996)



Christophe Letellier, G. Gouesbet. Topological Characterization of Reconstructed Attractors Modding Out Symmetries. Journal de Physique II, 1996, 6 (11), pp.1615-1638. ⟨10.1051/jp2:1996152⟩. ⟨jpa-00248397⟩
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