telech

2024

23/01/2024

Caterina Mosto, Gisela D. Charó, Christophe Letellier, & Denisse Sciamarella
Templex-based dynamical units for a taxonomy of chaos
Chaos, to appear.

Abstract
Discriminating different types of chaos is still a very challenging topic, even for dissipative three-dimensional systems for which the most advanced tool is the template. Nevertheless, getting a template is, by definition, limited to three-dimensional objects, since based on knot theory. To deal with higher-dimensional chaos, we recently introduced the templex combining a flow-oriented BraMAH cell complex and a directed graph (a digraph). There is no dimensional limitation in the concept of templex. Here, we show that a templex can be automatically reduced into a “minimal” form to provide a comprehensive and synthetic view of the main properties of chaotic attractors. This reduction allows for the development of a taxonomy of chaos in terms of two elementary units : the oscillating unit (O-unit) and the switching unit (S-unit). We apply this approach to various well-known attractors (Rössler, Lorenz, and Burke-Shaw) as well as a non-trivial four-dimensional attractor. A case of toroidal chaos (Deng) is also treated. This work is dedicated to Otto E. Rössler.

Christophe Letellier, Ludovico Minati, Irene Sendina-Nadal & I. Leyva
A taxonomy for generalized synchronization between flat-coupled systems
ArXiv, preprint submitted, 2024 Online

Abstract
Generalized synchronization is plausibly the most complex form of synchronization. Previous studies have revealed the existence of weak or strong forms of generalized synchronization depending on the multi- or mono-valued nature of the mapping between the attractors of two unidirectionally-coupled systems. Generalized synchronization is here obtained by coupling two systems with a flat control law. Here, we demonstrate that the corresponding first-return maps can be topologically conjugate in some cases. Conversely, the response map can foliated while the drive map is not. We describe the corresponding types of generalized synchronization, explicitly focusing on the influence of the coupling strength when significantly different dimensions or dissipation properties characterize the coupled systems. A taxonomy of generalized synchronization based on these properties is proposed.

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