I. Sendińa-Nadal & C. Letellier
Observability analysis and observer design of networks of Rössler systems
Chaos, 32 (8), 083109, 2022. Online
We address the problem of retrieving the full state of a network of Rössler systems from the knowledge of the actual state of a limited set of nodes. The selection of the nodes where sensors are placed is carried out in a hierarchical way through a procedure based on graphical and symbolic observability approaches applied to pairs of coupled dynamical systems. By using a map directly obtained from the governing equations, we design a nonlinear network reconstructor which is able to unfold the state of the non measured nodes with working accuracy. For sparse networks, the number of sensors scales with half the network size and node reconstruction errors are lower in networks with heterogeneous degree distributions. The method performs well even in the presence of parameter mismatch and non-coherent dynamics, and for dynamical systems with completely different algebraic structure like the Hindmarsch-Rose, therefore, we expect it to be useful for designing robust network control laws.
G. D. Charó, C. Letellier & D. Sciamarella
Templex : a bridge between homologies and templates for chaotic attractors
Chaos, 32 (8), 083108, 2022. Online
The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies : this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated — namely the spiral and funnel Rössler attractors, the Lorenz attractor, the Burke and Shaw attractor and a four-dimensional system. A link is established with their description in terms of templates.