Christophe Letellier

Université de Rouen | UR · Physics

This text is a collaborative work (14 co-authors) reviewing the development of nonlinear science in France. It is Chapt. 15 of the book "Les 150 ans de le société française de physique, panorama historique et scientifique", ISBN 978-2-7598-3075-6 under copyright EDP Science, SFP, 2023. Paul Manneville.

There is a tremendous fascination with chaos and fractals, about which picture books can be found on coffee tables everywhere. Chaos and fractals represent hands-on mathematics that is alive and changing. One can turn on a personal computer and create stunning mathematical images that no one has ever seen before. Chaos and fractals are part of dyna...

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 ter...

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 a...

Controlling the dynamics of chaotic systems is a task which is often addressed in an empirical way, particularly for placing sensors and actuators. Here, we show that selecting the measured variable and placing the actuator can be guided by considering the observability and controllability symbolic coefficients and applying the notion of flatness....

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 conditi...

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 healt...

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 propos...

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 topol...

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...

BACKGROUND Older adults suffering from cancer receive a treatment that, in most cases, causes symptoms such as fatigue, weakness, sleep disturbances, and lethargy. Moreover, the toxicity of the chemotherapy itself can also affect the physical function and the quality of life of cancer patients. Since a poor general health status reduces the chances...

We address the problem of retrieving the full state of a network of R\"ossler 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. By using a map directly obtaine...

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...

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 (exten...

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...

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 investigate...

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...

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 de...

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...

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 construct...

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...

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 attracto...

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 obse...

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 obse...

The design principle underlying Fig. 1.1 is easy to interpret in terms of realistic systems. The principle reads: ‘combine an oscillator with a switch and you are likely to obtain’.

Simple systems can produce complicated behavior. The reason lies in the structure of space and time. Simple ‘weaving rules’ suffice to determine very complicated motions.

The phenomenon of turbulence is, in a sense, too beautiful to be explained. It not only has infinite complexity—which is something we are accustomed to by now—but is non-repetitive in a much more creative way: it ‘draws’ a whole new flow pattern all the time.

If the presumable chaotic regime is low-dimensional and simple, the easiest thing to do is to prepare a next-amplitude map or some other one-dimensional projection of a cross section.

If complicated motions indeed occur so easily as has been suggest above, chaos should be among the ubiquitous phenomena of the world. This hypothesis seems not at variance with reality. To start out with a ‘far-fetched’ example, nonperiodically oscillating astronomical objects might be mentioned. Here the class of Cepheid pulsating stars is especia...

Lefschetz once remarked that Poincaré, after having discovered the possibility of homoclinic points, was well aware that from now on, the majority of simple dynamical systems could be expected to show the same complexity of behavior [1].

So far, a single design principle has proved sufficient to generate a whole world of nontrivial flows in three and higher dimensions. The resulting limiting equations appeared rather ‘robust’ in the sense that the limiting parameter \(\varepsilon \) could be increased from zero up to the order of unity (cf. Eq. ( 2.2)); and also in view of the fact...

As mentioned at the beginning, the reinjection is not confined to planar sub-systems. Reinjection processes between three-dimensional sub-systems, initiated along two-dimensional cross sections, are still quite easy to conceptualize and to generalize.

The oldest example of complicated behavior found in an unconstrained (non-conservative) dynamical system is, as mentioned in the Introduction, that found in the periodically forced van der Pol oscillator [1]. Cartwright and Littlewood’s results were based on painstaking analytical arguments (see the elaboration in [2]). Is it, at the same time, als...

Chaotic systems produce a characteristic noise, as described in Sect. 10.6. Does it suffice to diagnose chaos and, perhaps, also its higher forms?

Anaxagoras (see Introduction) inaugurated the philosophical hypothesis that a perfect mixture, existing since eons, could be unmixed. By adding the word ‘again’, we arrive at a mathematical version of the same conjecture. The deterministic maps of the preceding two sections are candidates examples. Even in its mathematical version, the conjecture h...

It would be astonishing if the just described ‘royal way’ to chaos was the only one. And it would also be astonishing if the next alternative—Lorenzian chaos—would not at the same time reveal a second principle — a whole second world.

Written in the 1980s by one of the fathers of chaos theory, Otto E. Rössler, the manuscript presented in this volume eventually never got published. Almost 40 years later, it remains astonishingly at the forefront of knowledge about chaos theory and many of the examples discussed have never been published elsewhere. The manuscript has now been edi...

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...

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 tim...

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 derive...

A dynamical network, a graph whose nodes are dynamical systems, is usually characterized by a large dimensional space which is not always accesible due to the impossibility of measuring all the variables spanning the state space. Therefore, it is of the utmost importance to determine a reduced set of variables providing all the required information...

We propose a new 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 and the lack of structure in the Poincar\'e plane constructed from a given time series. As for the former, we...

A dynamical network, a graph whose nodes are dynamical systems, is usually characterized by a large dimensional space which is not always accessible due to the impossibility of measuring all the variables spanning the state space. Therefore, it is of the utmost importance to determine a reduced set of variables providing all the required informatio...

Symptom monitoring during chemotherapy via web-based patient-reported outcomes (PROs) was previously demonstrated to lengthen survival in a single-center study.¹ A multicenter randomized clinical trial compared web-based monitoring vs standard scheduled imaging to detect symptomatic recurrence in patients with lung cancer following initial treatmen...

One of the main tasks in network theory is to infer relations among interacting elements. We propose global modeling as a tool to detect links between nodes and their nature. Various situations using small network motifs are investigated under the assumption that the variable to be measured at each node provides full observability when isolated. Su...

Two randomized studies of symptom monitoring during chemotherapy or during second line treatment and follow-up via web-based patient-reported outcomes (PROs) was previously demonstrated to lengthen survival. We are presenting here a patient with advanced and recurrent lung cancer who was followed for 4 years by PROs for relapse and adverse events d...

The basic laws of deterministic many-body systems are summarized in the footsteps of the deterministic approach pioneered by Yakov Sinai. Two fundamental cases, repulsive and attractive, are distinguished. To facilitate comparison, long-range potentials are assumed both in the repulsive case and in the new attractive case. In Part I, thermodynamics...

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 non...

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 ant...

The concept of observability of linear systems initiated with Kalman in the mid 1950s. Roughly a decade later, the observability of nonlinear systems appeared. By such definitions a system is either observable or not. Continuous measures of observability for linear systems were proposed in the 1970s and two decades ago were adapted to deal with non...

Dans le cas du handicap visuel, les expériences sociales passent par d'autres modalités sensorielles que la vision. Le but de notre étude est d'observer si une plus grande expérience du toucher influence la co-construction de l'interaction dans une situation de collaboration. Ainsi nous avons observé 58 participants (adolescents voyants et adolesce...

Model validation from experimental data is an important and not trivial topic which is too often reduced to a simple visual inspection of the state portrait spanned by the variables of the system. Synchronization was suggested as a possible technique for model validation. By means of a topological analysis, we revisited this concept with the help o...

A faithful description of the state of a complex dynamical network would require, in principle, the measurement of all its $d$ variables, an infeasible task for systems with practical limited access and composed of many nodes with high dimensional dynamics. However, even if the network dynamics is observable from a reduced set of measured variables...

When the state of the whole reaction network can be inferred by just measuring the dynamics of a limited set of nodes the system is said to be fully observable. However, as the number of all possible combinations of measured variables and time derivatives spanning the reconstructed state of the system exponentially increases with its dimension, the...

Résumé Dans le cas du handicap visuel, les expériences sociales passent par d’autres modalités sensorielles que la vision. Le but de notre étude est d’observer si une plus grande expérience du toucher influence la co-construction de l’interaction dans une situation de collaboration. Ainsi nous avons observé 58 participants (adolescents voyants et a...

Synchronization is a very generic process commonly observed in a large variety of dynamical systems which, however, has been rarely addressed in systems with low dissipation. Using the Rössler, the Lorenz 84, and the Sprott A systems as paradigmatic examples of strongly, weakly, and non-dissipative chaotic systems, respectively, we show that a para...

In this talk various aspects of observability will be reviewed. The aim is to make a clear distinction between concepts and to understand what does each one contribute to the analysis and monitoring of networks. A dynamical network will be considered. Two sets of simulations were performed. In one we test for the observability of the graph. In th s...

It is well known that cancers are significantly more often encountered in some tissues than in other ones. In this paper, by using a deterministic model describing the interactions between host, effector immune and tumor cells at the tissue level, we show that this can be explained by the dependency of tumor growth on parameter values characterizin...

The genesis of special relativity is intimately related to the development of the theory of light propagation. When optical phenomena were described, there are typically two kinds of theories: (i) One based on light rays and light particles and (ii) one considering the light as waves. When diffraction and refraction were experimentally discovered,...

Intermittencies are commonly observed in fluid mechanics, and particularly, in pipe flows. Initially observed by Reynolds (1883), it took one century for reaching a rather full understanding of this phenomenon whose irregular dynamics (apparently stochastic) puzzled hydrodynamicists for decades. In this brief (non-exhaustive) review, mostly focused...

Combined therapy made of a chemotherapy and antiangiogenic agents is a clinical treatment recommended for its efficiency. Since the optimization of a treatment against cancer relasp is still mostly based on oncologist’s know-how, it is desirable to develop different approaches for such a task. Mathematical modelling is one of the promising ways. We...

Observability is the property that enables to distinguish two different locations in $n$-dimensional state space from a reduced number of measured variables, usually just one. In high-dimensional systems it is therefore important to make sure that the variable recorded to perform the analysis conveys good observability of the system dynamics. In th...

The genesis of special relativity is intimately related to the development of the theory of light propagation. When optical phenomena were described, there are typically two kinds of theories: (i) One based on light rays and light particles and (ii) one considering the light as waves. When diffraction and refraction were experimentally discovered,...

If recent advances in oncology emphasized the role of microenvironment in tumor growth, the role of delays for modeling tumor growth is still uncertain. In this paper, we considered a model, describing the interactions of tumor cells with their microenvironment made of immune cells and host cells, in which we inserted, as suggested by the clinician...

Background: The use of web-based monitoring for lung cancer patients is growing in interest because of promising recent results suggesting improvement in cancer and resource utilization outcomes. It remains an open question whether the overall survival (OS) in these patients could be improved by using a web-mediated follow-up rather than classical...

Les fluides physiologiques servent au transport de molécules impliqués dans le fonctionnement des grands organismes. Par exemple, l’air est apporté aux poumons par la ventilation, le sang diffuse l’oxygène jusqu’aux muscles et extrait les « déchets » comme le dioxyde de carbone pour les éliminer du corps. Ce livre se présente comme une introduction...

In the case of visual impairment, social experiences are through other sensory modalities than vision. The aim of our study is to observe whether a greater experience of touch influences the co-construction of interaction in a collaborative task. We observed 58 participants (visually impaired tennagers and sighted teenagers) during a situation of c...

Background Pulmonary arterial hypertension is a rare and lethal disease affecting small diameter pulmonary arteries and leading to a progressive increase of the right vascular resistances. Patients with such a disease have no specific symptom, a feature which delays the diagnosis by 18 months to 2 years in average. In most cases, pulmonary arterial...

Background Noninvasive mechanical ventilation is sufficiently widely used to motivate bench studies for evaluating and comparing performances of the domiciliary ventilators. In most (if not in all) of the previous studies, ventilators were tested in a single (or a very few) conditions, chosen to avoid asynchrony events. Such a practice does not ref...

Describing tumor growth is a key issue in oncology for correctly understanding the underlying mechanisms leading to deleterious cancers. In order to take into account the micro-environment in tumor growth, we used a model describing — at the tissue level — the interactions between host (non malignant), effector immune and tumor cells to simulate th...

Understanding the conditions under which a collective dynamics emerges in a complex network is still an open problem. A useful approach is the master stability function—and its related classes of synchronization—which offers a necessary condition to assess when a network successfully synchronizes. Observability coefficients, on the other hand, quan...

Background: The use of web-based monitoring for lung cancer patients is growing in interest because of promising recent results suggesting improvement in cancer and resource utilization outcomes. It remains an open question whether the overall survival (OS) in these patients could be improved by using a web-mediated follow-up rather than classical...

Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space remains an open question. We here want to explain—in the particular case of the Wei system—such a structure, using one-dimensional sets obtained by vanishing two of th...

In the two last decades the concept of observability has been formally linked to that of embedding in the context of nonlinear dynamics. Such a concept has been shown to play an important role in global modeling, data analysis and filtering, to mention a few exam- ples. Preliminary results suggested that observability, at least in some cases, has s...

Spatially distributed systems are rather difficult to investigate due to two distinct problems which can be sometimes combined. First, the spatial extension is taken into account by monitoring the system evolution at different locations. Second, the dynamics cannot always be continuously tracked in time, and segments of data – sometimes recorded at...

Objectifs La détection précoce des rechutes après chimioradiothérapie ou chimiothérapie des carcinomes bronchiques est un enjeu important. Un précédent essai de phase 2 non randomisé a montré que l’autosurveillance à l’aide de la web-application « sentinel » pouvant déclencher une alerte de récidive auprès de l’oncologue avait possiblement un effet...

A suspension of a map consists of the flow for which the Poincaré section is that map. Designing a suspension of a given map remains a non-trivial task in general. The case of suspending the Hénon map is here considered. Depending on the parameter values, the Hénon map is orientation preserving or reversing; it is here shown that while a tridimensi...

The genesis of special relativity is intimately related to the development of the theory of light propagation. When optical phenomena were described, there are typically two kinds of theories: (i) One based on light rays and light particles and (ii) one considering the light as waves. When diffraction and refraction were experimentally discovered,...

Heart rate variability analysis using 24-h Holter monitoring is frequently performed to assess the cardiovascular status of a patient. The present retrospective study is based on the beat-to-beat interval variations or ΔRR, which offer a better view of the underlying structures governing the cardiodynamics than the common RR-intervals. By investiga...