S. Mangiarotti & C. Letellier
Topological characterization of toroidal chaos : A branched manifold for the Deng toroidal attractor
Chaos, 31, 013129, 2021. Online
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 Rössler 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, R. Abraham, D. L. Shepelyansky, O. E. Rössler, P. Holmes, R. Lozi, L. Glass, A. Pikovsky, L. F. Olsen, I. Tsuda, C. Grebogi, U. Parlitz, R. Gilmore, L. M. Pecora, & T. L. Carroll
Some elements for a history of the dynamical systems theory
Chaos, 31, 053110 (2021) ; Online
Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to “reconstruct” some supposed influences. In the 1970s emerged a new way of performing science under the name “chaos”, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements — which were never published — illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.
Branched manifolds for the three types of unimodal maps
Communications in Nonlinear Science and Numerical Simulation, 101, 105869, 2021. Online
Branched manifold is certainly the finest description of the structure of chaotic attractors, characterizing how the unstable periodic orbits are knotted. Many chaotic attractors produced by strongly dissipative systems were thus topologically described. In spite of this, the different possibilities for the branched manifolds which may be constructed from unimodal maps were never exhaustively listed. This is the task of the present work, starting from the folded (Logistic) map, the torn (Lorenz) map, and the less known torn away map introduced by Rössler in 1979. The case of the “reverse” horseshoe map is also discussed.
C. Letellier, I. Sendiña-Nadal, L. Minati & I. Leyva
Node differentiation dynamics along the route to synchronization in complex networks
Physical Review E, 104, 014303, 2021. Online
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.
C. Letellier, M. Lujan, J.-M. Arnal, A. Carlucci, M. Chatwin, B. Ergan, M. Kampelmacher, J. H. Storre, N. Hart, J. Gonzalez-Bermejo & S. Nava
Patient-ventilator synchronization during non-invasive ventilation : a pilot study of an automated analysis system
Frontiers in Medical Technology. Diagnostic and Therapeutic Devices, 3, 690442, 2021. Online
BACKGROUND : Patient-ventilator synchronization during noninvasive ventilation (NIV) can be assessed by visual inspection of flow and pressure waveforms but it remains time consuming and there is a large inter-rater variability, even among expert physicians. SyncSmart software developed by Breas Medical (Mölnycke, Sweden) provides an automatic detection and scoring of patient-ventilator asynchrony to help physicians in their daily clinical practice. This study was designed to assess performance of the automatic scoring by the SyncSmart software using expert clinicians as a reference in patient with chronic respiratory failure receiving NIV.
METHODS : From 9 patients, 20 min data sets were analyzed automatically by SyncSmart software and reviewed by 9 expert physicians who were asked to score auto-triggering (AT), double-triggering (DT), and ineffective efforts (IE). The study procedure was similar to the one commonly used for validating the automatic sleep scoring technique. For each patient, the asynchrony index was computed by automatic scoring and each expert, respectively. Considering successively each expert scoring as a reference, sensitivity, specificity, positive predictive value (PPV), κ-coefficients, and agreement were calculated.
RESULTS : The asynchrony index assessed by SyncSmart was not significantly different from the one assessed by the experts (18.9 ± 17.7 versus 12.8 ± 9.4, p = 0.19). When compared to an expert, the sensitivity and specificity provided by SyncSmart for DT, AT, and IE were significantly greater than those provided by an expert when compared to another expert.
CONCLUSIONS : SyncSmart software is able to score asynchrony events within the inter-rater variability. When the breathing frequency is not too high (< 24 bpm), it therefore provides a reliable assessment of patient-ventilator asynchrony ; AT is over detected otherwise.
The datasets are available Here.
C. Letellier, L. F. Olsen & Sylvain Mangiarotti
Chaos : From theory to applications for the 80th birthday of Otto E. Rössler
Chaos 31, 060402, 2021. Online
We took the opportunity of the 80th birthday of Rössler (born on May 20, 1940) to acknowledge him this Focus Issue for his contribution to the chaos theory. Our purpose was to promote chaos, its inherent properties, as well as its applications. It consists of 18 papers which are briefly introduced in the subsequent part of this introduction.
E M. A. M. Mendes, C. Lainscsek & C. Letellier
Diffeomorphical equivalence versus Topological equivalence among Sprott systems
Chaos, 31, 083126, 2021. Online
In 1994, Sprott  proposed a set of 19 different simple dynamical systems produc- ing chaotic attractors. Among them, 14 systems have a single nonlinear term. To the best of our knowledge, their diffeomorphical equivalence and the topological equivalence of their chaotic attractors were never systematically in- vestigated. This is the aim of this paper. We here propose to check their diffeomorphical equivalence through the jerk functions which are obtained when the system is rewritten in terms of one of the variables and its first two derivatives (two systems are thus diffeomorphically equivalent when they have exactly the same jerk function, that is, the same functional form and the same coefficients). The chaotic attractors produced by these systems — for parameter values close to the ones initially proposed by Sprott — are characterized by a branched manifold. Systems B and C produce chaotic attractors, which are observed in the Lorenz system, are also briefly discussed. Those systems are classified according to their diffeomorphical and topological equivalence.
C. Letellier, N. Stankevich & O. E. Rössler
Dynamical Taxonomy : some taxonomic ranks to systematically classify every chaotic attractor
International Journal of Bifurcation & Chaos, accepted. ArXiv
Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine what are the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labelling. Addressing these problems correspond 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, covering a large variety of known (and less known) examples of chaotic systems. 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 taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multi-component attractors (bounded by a torus whose genus g <= 3
C. Letellier & J.-P. Barbot
Optimal flatness placement of sensors and actuators for controlling chaotic systems
Controlling chaotic systems is very often investigated by using empirical laws, without taking advantage ofnthe structure of the governing equations. There are two concepts, observability and controllability, whichnare inherited from control theory, for selecting the best placement of sensors and actuators. These twonconcepts can be combined (extended) into flatness which provides the conditions to fulfill for designing anfeedback linearization or another classical control law for which the system is always fully observable and fullyncontrollable. We here design feedback linearization control laws using flatness for the three popular chaoticnsystems, namely, the Rössler, the driven van der Pol and the Hénon-Heiles systems. As developed duringnthe last two decades for observability, symbolic controllability coefficients and symbolic flatness coefficients are here introduced and their meanings are tested with numerical simulations. We show that the control lawnworks for every initial conditions when the symbolic flatness coefficient is equal to 1.