Topological characterization and reconstruction of strange attractors
CARACTERISATION TOPOLOGIQUE et RECONSTRUCTION D'ATTRACTEURS ETRANGES
Résumé
Chaos theory can be applied to various domains. But in order to do that, there is a great demand to possess a better description of chaotic attractors as well as the underlying mechanisms responsible for these new types of behaviors. Such a task requires a quite theoretical approach. Roughly, two types of techniques are investigated in this Ph’D thesis: i) topological characterization and ii) global modelling. The first deals with the relative organization of unstable periodic orbits around which a chaotic attractor is structured. The original contribution of this work is to develop an appropriate way to investigate systems with symmetry property as the Lorenz system has. In particular, it is shown that there is a great advantage to modd out the symmetry property to correctly identify the relevant dynamical properties of equivariant dynamical systems. The second type technique concerns the obtention of a set of ordinary differential equations from experimental data. It is shown that it is possible to recover a set of differential equations estimated from the experimental dynamics reconstructed in a space spanned by derivative coordinates. In particular the case of noise contaminated data is solved using a smoothing procedure prior to any attempt for global modeling. In the third part of this thesis, few examples from electrochemistry and astrophysics are are investigated.
Chaos theory can be applied to various domains. But in order to do that, there is a great demand to possess a better description of chaotic attractors as well as the underlying mechanisms responsible for these new types of behaviors. Such a task requires a quite theoretical approach. Roughly, two types of techniques are investigated in this Ph’D thesis: i) topological characterization and ii) global modelling. The first deals with the relative organization of unstable periodic orbits around which a chaotic attractor is structured. The original contribution of this work is to develop an appropriate way to investigate systems with symmetry property as the Lorenz system has. In particular, it is shown that there is a great advantage to modd out the symmetry property to correctly identify the relevant dynamical properties of equivariant dynamical systems. The second type technique concerns the obtention of a set of ordinary differential equations from experimental data. It is shown that it is possible to recover a set of differential equations estimated from the experimental dynamics reconstructed in a space spanned by derivative coordinates. In particular the case of noise contaminated data is solved using a smoothing procedure prior to any attempt for global modeling. In the third part of this thesis, few examples from electrochemistry and astrophysics are are investigated.
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