S. Mangiarotti & C. Letellier
Topological characterization of toroidal chaos : A branched manifold for the Deng toroidal attractor
Chaos, 31, 013129, 2021. Online

- Abstract
When a chaotic attractor is produced by a three-dimensional strongly dissipative system, its ultimate characterization is reached when a branched manifold—a template—can be used to describe the relative organization of the unstable periodic orbits around which it is structured. If topological characterization was completed for many chaotic attractors, the case of toroidal chaos—a chaotic regime based on a toroidal structure—is still challenging. We here investigate the topology of toroidal chaos, first by using an inductive approach, starting from the branched manifold for the R÷ssler attractor. The driven van der Pol system—in Robert Shaw’s form—is used as a realization of that branched manifold. Then, using a deductive approach, the branched manifold for the chaotic attractor produced by the Deng toroidal system is extracted from data.

C. Letellier, I. Sendi˝a-Nadal, L. Minati & I. Leyva
Node differentiation dynamics along the route to synchronization in complex networks
ArXiv, 2102.09989, 2021. Online

- Abstract
Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper, we show how in a network of identical dynamical systems, nodes belonging to the same degree class differentiate in the same manner visiting a sequence of states of diverse complexity along the route to synchronization independently on the global network structure. In particular, we observe, just after interaction starts pulling orbits from the initially uncoupled attractor, a general reduction of the complexity of the dynamics of all units being more pronounced in those with higher connectivity. In the weak coupling regime, when synchronization starts to build up, there is an increase in the dynamical complexity whose maximum is achieved, in general, first in the hubs due to their earlier synchronization with the mean field. For very strong coupling, just before complete synchronization, we found a hierarchical dynamical differentiation with lower degree nodes being the ones exhibiting the largest complexity departure. We unveil how this differentiation route holds for several models of nonlinear dynamics including toroidal chaos and how it depends on the coupling function. This study provides new insights to understand better strategies for network identification and control or to devise effective methods for network inference.

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