Christophe LETELLIER
12/06/2023

Gisela Charo, Christophe Letellier & Denisse Sciamarella

Developing methods to accomplish a topological description of the structure of a flow in high-dimensional state space (in more than three dimensions) has involved several false starts as well as partially successful roads, that finally lead to what we now call a templex. [1] Cell complexes can be traced back to Poincaré’s papers in the late 1800s [2] and the study of chaotic attractors using cell complexes to the 1990s [3] [4] Since then, algebraic topology has been seen as the mathematical formalism holding promise for a description of chaos beyond three dimensions --- there where templates, developed in the 1980s to extract the knot content of attractors, cannot go. In this talk, we present the road leading to the templex starting from the Rössler attractor [5] and ending with a four-dimensional system designed in Ref. [4] on the basis of a set of equations proposed by Deng [6]

Voir ce site : Denisse Sciamarella

[1] **G. D. Charó, C. Letellier, & D. Sciamarella**, Templex : A bridge between homologies and templates for chaotic attractors. *Chaos*, **32** (8), 083108, 2022.

[2] **H. Poincaré**, Analysis situs, *Journal de l’École Polytechnique*, **1**, 1-121, 1895.

[3] **D. Sciamarella & G. B. Mindlin**, Topological structure of chaotic flows from human speech data. *Physical review Letters*, **82** (7), 1450, 1999.

[4] **D. Sciamarella, & G. B. Mindlin**, Unveiling the topological structure of chaotic flows from data. *Physical review E*, **64** (3), 036209, 2001.

[5] **O. E. Rössler**, An equation for continuous chaos, *Physics Letters A*, **57**, 397-398, 1976.

[6] **B. Deng**, Constructing homoclinic orbits and chaotic attractors, *International Journal of Bifurcation & Chaos*, **4** (4), 823-841, 1994.