Christophe LETELLIER
15/03/2022

I. Sendińa-Nadal & C. Letellier

Observability analysis and state reconstruction for networks of nonlinear systemsChaos,32(8), 083109, 2022. Online

Abstract

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 attractorsChaos,32(8), 083108, 2022. Online

Abstract

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.

L. A. Aguirre, F. B. Freitas & C. Letellier

Numerical interpretation of controllability coefficients in nonlinear dynamicsCommunications in Nonlinear Science and Numerical Simulation,116, 106875, 2023. Online

Abstract

As its dual, observability, the controllability of controllable systems actuated in different ways can be quantified by coefficients which can be computed either by a numerical approach or a symbolic one. If the interpretation of observability coefficients is rather straightforward, this is not the case for controllability. This paper, after proposing slight but important adjustments to the computation of numerical controllability coefficients puts forward a numerical framework that can be used to provide some clue as how to interpret the role played by numerical and symbolical controllability coefficients. Using four different chaotic systems, it is showed how controllability may depend on the derivative to which the control law is applied and how this may be related to the corresponding coefficients. In practice could help to decide where to place actuators. The developed framework has two backbones : the power needed to achieve control and the rate of success of the control, the latter is not considered in the surveyed literature. Our results show that numerical and symbolic coefficients quantify different aspects of the problem and confirm how controllability is difficult to interpret in practice without any consideration for observability.

C. Letellier, N. Stankevich & O. E. Rössler

Dynamical taxonomy : Some axotnomic ranks to systematically classify every chaotic attractorInternational Journal of Bifurcation & Chaos,32(2), 2230004, 2022. Online

Abstract

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaos. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map, . . . ).By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus

ggreater than 2).