Lorenz-like systems and Lorenz-like attractors: Definition, examples, and equivalences.

Since the early 1970s, numerous systems exhibiting an algebraic structure resembling that of the 1963 Lorenz system have been proposed. These systems have occasionally yielded the same attractor as the Lorenz system, while in other cases, they have not. Conversely, some systems that are evidently distinct from the Lorenz system, particularly in terms of symmetry, have resulted in attractors that bear a resemblance to the Lorenz attractor. In this paper, we put forward a definition for Lorenz-like systems and Lorenz-like attractors. The former definition is based on the algebraic structure of the governing equations, while the latter relies on topological characterization. Our analysis encompasses over 20 explicitly examined chaotic systems.

Generalized synchronization mediated by a flat coupling between structurally nonequivalent chaotic systems.

Synchronization of chaotic systems is usually investigated for structurally equivalent systems typically coupled through linear diffusive functions. Here, we focus on a particular type of coupling borrowed from a nonlinear control theory and based on the optimal placement of a sensor-a device measuring the chosen variable-and an actuator-a device applying the actuating (control) signal to a variable's derivative-in the response system, leading to the so-called flat control law. We aim to investigate the dynamics produced by a response system that is flat coupled to a drive system and to determine the degree of generalized synchronization between them using statistical and topological arguments. The general use of a flat control law for getting generalized synchronization is discussed.

Optimal placement of sensor and actuator for controlling low-dimensional chaotic systems based on global modeling.

Controlling chaos is fundamental in many applications, and for this reason, many techniques have been proposed to address this problem. Here, we propose a strategy based on an optimal placement of the sensor and actuator providing global observability of the state space and global controllability to any desired state. The first of these two conditions enables the derivation of a model of the system by using a global modeling technique. In turn, this permits the use of feedback linearization for designing the control law based on the equations of the obtained model and providing a zero-flat system. The procedure is applied to three case studies, including two piecewise linear circuits, namely, the Carroll circuit and the Chua circuit whose governing equations are approximated by a continuous global model. The sensitivity of the procedure to the time constant of the dynamics is also discussed.

Physical Activity Program for the Survival of Elderly Patients With Lymphoma: Study Protocol for Randomized Phase 3 Trial.

BACKGROUND: The practice of regular physical activity can reduce the incidence of certain cancers (colon, breast, and prostate) and improve overall survival after treatment by reducing fatigue and the risk of relapse. This impact on survival has only been demonstrated in active patients with lymphoma before and after treatment. As poor general health status reduces the chances of survival and these patients are most likely to also have sarcopenia, it is important to be able to improve their physical function through adapted physical activity (APA) as part of supportive care management. Unfortunately, APA is often saved for patients with advanced blood cancer. As a result, there is a lack of data regarding the impact of standardized regular practice of APA and concomitant chemotherapy as first-line treatment on lymphoma survival. OBJECTIVE: This study aimed to assess the impact of a new and open rehabilitation program suitable for a frail population of patients treated for diffuse large B-cell lymphoma (DLBCL). METHODS: PHARAOM (Physical Activity Program for the Survival of Elderly Patients with Lymphoma) is a phase 3 randomized (1:1) study focusing on a frail population of patients treated for DLBCL. The study will include 186 older adult patients with DLBCL (aged >65 years) receiving rituximab and chemotherapy. Overall, 50% (93/186) of patients (investigational group) will receive APA along with chemotherapy, and they will be supervised by a dedicated qualified kinesiologist. The APA program will include endurance and resistance training at moderate intensity 3 times a week during the 6 months of chemotherapy. The primary end point of this study will be event-free survival of the patients. The secondary end points will include the overall survival, progression-free survival, prevalence of sarcopenia and undernutrition, and patients' quality of life. This study will be conducted in accordance with the principles of the Declaration of Helsinki. RESULTS: Recruitment, enrollment, and data collection began in February 2021, and 4 participants have been enrolled in the study as of July 2022. Data analysis will begin after the completion of data collection. Future outcomes will be published in peer-reviewed health-related research journals and presented at national congress, and state professional meetings. This publication is based on protocol version 1.1, August 3, 2020. CONCLUSIONS: The PHARAOM study focuses on highlighting the benefits of APA intervention on survival during the period of first-line treatment of patients with DLBCL. This study could also contribute to our understanding of how an APA program can reduce complications such as sarcopenia in patients with lymphoma and improve their quality of life. By documenting the prevalence and relationship between sarcopenia and exercise load, we might be able to help physicians plan better interventions in the care of patients with DLBCL. TRIAL REGISTRATION: ClinicalTrials.gov NCT04670029; https://clinicaltrials.gov/ct2/show/NCT04670029. INTERNATIONAL REGISTERED REPORT IDENTIFIER (IRRID): DERR1-10.2196/40969.

Observability analysis and state reconstruction for networks of nonlinear systems.

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 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 governing equations, we design a nonlinear network reconstructor that is able to unfold the state of non-measured nodes with working accuracy. For sparse networks, the number of sensor 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 structures like the Hindmarsch-Rose; therefore, we expect it to be useful for designing robust network control laws.

Templex: A bridge between homologies and templates for chaotic attractors.

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.

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 of the structure of the governing equations. There are two concepts, observability and controllability, which are inherited from control theory, for selecting the best placement of sensors and actuators. These two concepts can be combined (extended) into flatness, which provides the conditions to fulfill for designing a feedback linearization or another classical control law for which the system is always fully observable and fully controllable. We here design feedback linearization control laws using flatness for the three popular chaotic systems, namely, the Rössler, the driven van der Pol, and the Hénon-Heiles systems. As developed during the last two decades for observability, symbolic controllability coefficients and symbolic flatness coefficients are introduced here and their meanings are tested with numerical simulations. We show that the control law works for every initial condition when the symbolic flatness coefficient is equal to 1.

Diffeomorphical equivalence vs topological equivalence among Sprott systems.

In 1994, Sprott [Phys. Rev. E 50, 647-650 (1994)] proposed a set of 19 different simple dynamical systems producing 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 investigated. 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 and are also briefly discussed. Those systems are classified according to their diffeomorphical and topological equivalence.

Node differentiation dynamics along the route to synchronization in complex networks.

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 insights to understand better strategies for network identification or to devise effective methods for network inference.

Some elements for a history of the dynamical systems theory.

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, a new way of performing science under the name "chaos" emerged, 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.

Topological characterization of toroidal chaos: A branched manifold for the Deng toroidal attractor.

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.

Patient-Ventilator Synchronization During Non-invasive Ventilation: A Pilot Study of an Automated Analysis System.

Background: Patient-ventilator synchronization during non-invasive 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 nine patients, 20 min data sets were analyzed automatically by SyncSmart software and reviewed by nine 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 SynSmart was not significantly different from the one assessed by the experts (18.9 ± 17.7 vs. 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), it therefore provides a reliable assessment of patient-ventilator asynchrony; AT is over detected otherwise.

Assessing observability of chaotic systems using Delay Differential Analysis.

Observability can determine which recorded variables of a given system are optimal for discriminating its different states. Quantifying observability requires knowledge of the equations governing the dynamics. These equations are often unknown when experimental data are considered. Consequently, we propose an approach for numerically assessing observability using Delay Differential Analysis (DDA). Given a time series, DDA uses a delay differential equation for approximating the measured data. The lower the least squares error between the predicted and recorded data, the higher the observability. We thus rank the variables of several chaotic systems according to their corresponding least square error to assess observability. The performance of our approach is evaluated by comparison with the ranking provided by the symbolic observability coefficients as well as with two other data-based approaches using reservoir computing and singular value decomposition of the reconstructed space. We investigate the robustness of our approach against noise contamination.

Assessing synchronizability provided by coupling variable from the algebraic structure of dynamical systems.

Synchronization is a very generic phenomenon which can be encountered in a large variety of coupled dynamical systems. Being able to synchronize chaotic systems is strongly dependent on the nature of their coupling. Few attempts to explain such a dependency using observability and/or controllability were not fully satisfactory and synchronizability yet remained unexplained. Synchronizability can be defined as the range of coupling parameter values for which two nearly identical systems are fully synchronized. Our objective is here to investigate whether synchronizability can be related to the main rotation necessarily required for structuring any type of attractor, that is, whether synchronizability is significantly improved when the coupling variable is one of the variables involved in the main rotation. We thus propose a semianalytic procedure from a single isolated system to discard the worst variable for fully synchronizing two (nearly) identical copies of that system.

Dynamical complexity measure to distinguish organized from disorganized dynamics.

We propose a metric to characterize the complex behavior of a dynamical system and to distinguish between organized and disorganized complexity. The approach combines two quantities that separately assess the degree of unpredictability of the dynamics and the lack of describability of the structure in the Poincaré plane constructed from a given time series. As for the former, we use the permutation entropy S_{p}, while for the latter, we introduce an indicator, the structurality Δ, which accounts for the fraction of visited points in the Poincaré plane. The complexity measure thus defined as the sum of those two components is validated by classifying in the (S_{p},Δ) space the complexity of several benchmark dissipative and conservative dynamical systems. As an application, we show how the metric can be used as a powerful biomarker for different cardiac pathologies and to distinguish the dynamical complexity of two electrochemical dissolutions.

Observability of laminar bidimensional fluid flows seen as autonomous chaotic systems.

Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Bénard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed.

Parameter identification of a model for prostate cancer treated by intermittent therapy.

Adenocarcinoma is the most frequent cancer affecting the prostate walnut-size gland in the male reproductive system. Such cancer may have a very slow progression or may be associated with a "dark prognosis" when tumor cells are spreading very quickly. Prostate cancers have the particular properties to be marked by the level of prostate specific antigen (PSA) in blood which allows to follow its evolution. At least in its first phase, prostate adenocarcinoma is most often hormone-dependent and, consequently, hormone therapy is a possible treatment. Since few years, hormone therapy started to be provided intermittently for improving patient's quality of life. Today, durations of on- and off-treatment periods are still chosen empirically, most likely explaining why there is no clear benefit from the survival point of view. We therefore developed a model for describing the interaction between the tumor environment, the PSA produced by hormone-dependent and hormone-independent tumor cells, respectively, and the level of androgens. Model parameters were identified using a genetic algorithm applied to the PSA time series measured in a few patients who initially received prostatectomy and were then treated by intermittent hormone therapy (LHRH analogs and anti-androgen). The measured PSA time series is quite correctly reproduced by free runs over the whole follow-up. Model parameter values allow for distinguishing different types of patient (age and Gleason score) meaning that the model can be individualized. We thus showed that the long-term evolution of the cancer can be affected by durations of on- and off-treatment periods.

Structural, dynamical and symbolic observability: From dynamical systems to networks.

Classical definitions of observability classify a system as either being observable or not. Observability has been recognized as an important feature to study complex networks, and as for dynamical systems the focus has been on determining conditions for a network to be observable. About twenty years ago continuous measures of observability for nonlinear dynamical systems started to be used. In this paper various aspects of observability that are established for dynamical systems will be investigated in the context of networks. In particular it will be discussed in which ways simple networks can be ranked in terms of observability using continuous measures of such a property. Also it is pointed out that the analysis of the network topology is typically not sufficient for observability purposes, since both the dynamics and the coupling of such nodes play a vital role. Some of the main ideas are illustrated by means of numerical simulations.