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.

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.

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.

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.

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.

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.

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.

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.

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.

Using global modeling to unveil hidden couplings in small network motifs.

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. Such a choice ensures no intrinsic difficulties for getting a global model in the coupled situation. As a first step toward unveiling the coupling function in larger network motifs, we consider three different scenarios involving Rössler systems diffusively coupled, in a couple or embedded in a network, or parametrically forced. We show that the global modeling is able to determine not only the existence of an interaction but also its functional form, to retrieve the dynamics of the whole system, and to extract the equations governing the single node dynamics as if it was isolated.

Topological characterization versus synchronization for assessing (or not) dynamical equivalence.

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 of an abstract chemical reaction system and data from two electrodissolution experiments conducted by Jack Hudson's group. The fact that it was possible to synchronize topologically different global models led us to conclude that synchronization is not a recommendable technique for model validation. A short historical preamble evokes Jack Hudson's early career in interaction with Otto E. Rössler.

Spatial avascular growth of tumor in a homogeneous environment.

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 the evolution of cancer. The spatial growth is described by a Laplacian operator for the diffusion of tumor cells. We investigated how the evolution of the tumor diameter is related to the dynamics (periodic or chaotic oscillations, stable singular points) underlying the interactions between the different populations of cells in proliferation sites. The sensitivity of this evolution to the key parameter responsible for the immuno-evasion, namely the growth rate of effector immune cells and their inhibition rate by tumor cells, is also investigated.

Synchronizability of nonidentical weakly dissipative systems.

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 parameter or frequency mismatch between two coupled such systems does not affect the synchronizability and the underlying structure of the joint attractor in the same way. By computing the Shannon entropy associated with the corresponding recurrence plots, we were able to characterize how two coupled nonidentical chaotic oscillators organize their dynamics in different dissipation regimes. While for strongly dissipative systems, the resulting dynamics exhibits a Shannon entropy value compatible with the one having an average parameter mismatch, for weak dissipation synchronization dynamics corresponds to a more complex behavior with higher values of the Shannon entropy. In comparison, conservative dynamics leads to a less rich picture, providing either similar chaotic dynamics or oversimplified periodic ones.

How the growth rate of host cells affects cancer risk in a deterministic way.

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 characterizing the type as well as the state of the tissue considered due to the "way of life" (environmental factors, food consumption, drinking or smoking habits, etc.). Our approach is purely deterministic and, consequently, the strong correlation (r = 0.99) between the number of detectable growing tumors and the growth rate of cells from the nesting tissue can be explained without evoking random mutation arising during DNA replications in nonmalignant cells or "bad luck". Strategies to limit the mortality induced by cancer could therefore be well based on improving the way of life, that is, by better preserving the tissue where mutant cells randomly arise.

Observability and synchronization of neuron models.

Observability is the property that enables recovering the state of a dynamical system from a reduced number of measured variables. 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. The observability of a network of neuron models depends nontrivially on the observability of the node dynamics and on the topology of the network. The aim of this paper is twofold. First, to perform a study of observability using four well-known neuron models by computing three different observability coefficients. This not only clarifies observability properties of the models but also shows the limitations of applicability of each type of coefficients in the context of such models. Second, to study the emergence of phase synchronization in networks composed of neuron models. This is done performing multivariate singular spectrum analysis which, to the best of the authors' knowledge, has not been used in the context of networks of neuron models. It is shown that it is possible to detect phase synchronization: (i) without having to measure all the state variables, but only one (that provides greatest observability) from each node and (ii) without having to estimate the phase.

Architecture of chaotic attractors for flows in the absence of any singular point.

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 the three derivatives of the flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2 × 2 matrix describing the stability of flow in a subspace transverse to it. We will show that the attractor is spiralling and twisted in the neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.

An easy-to-use technique to characterize cardiodynamics from first-return maps on ΔRR-intervals.

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 investigating data for three groups of adults (with normal sinus rhythm, congestive heart failure, and atrial fibrillation, respectively), we showed that the first-return maps built on ΔRR can be classified according to three structures: (i) a moderate central disk, (ii) a reduced central disk with well-defined segments, and (iii) a large triangular shape. These three very different structures can be distinguished by computing a Shannon entropy based on a symbolic dynamics and an asymmetry coefficient, here introduced to quantify the balance between accelerations and decelerations in the cardiac rhythm. The probability P111111 of successive heart beats without large beat-to-beat fluctuations allows to assess the regularity of the cardiodynamics. A characteristic time scale, corresponding to the partition inducing the largest Shannon entropy, was also introduced to quantify the ability of the heart to modulate its rhythm: it was significantly different for the three structures of first-return maps. A blind validation was performed to validate the technique.