Observability coefficients for predicting the class of synchronizability from the algebraic structure of the local oscillators

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.

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.

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.

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.

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.

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.

Nonlinear graph-based theory for dynamical network observability

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 high dimensional systems due to practical limitations. However the network dynamics might be observable from a reduced set of measured variables but how to reliably identify the minimum set of variables providing full observability still remains an unsolved problem. In order to tackle this issue from the Jacobian matrix of the governing equations, we construct a pruned fluence graph in which the nodes are the state variables and the links represent only the linear dynamical interdependences after having ignored the nonlinear ones. From this graph, we identify the largest connected subgraphs with no outgoing links in which every node can be reached from any other node in the subgraph. In each one of them, at least one node must be measured to correctly monitor the state of the system in a d-dimensional reconstructed space. Our procedure is here tested by investigating large-dimensional reaction networks. Our results are validated by comparing them with the determinant of the observability matrix which provides a rigorous assessment of the system's observability.

Limits on reconstruction of dynamics in networks

An observability condition number is defined for physical systems modeled by network dynamics. Assuming that the dynamical equations of the network are known and a noisy trajectory is observed at a subset of the nodes, we calculate the expected distance to the nearest correct trajectory as a function of the observation noise level and discuss how it varies over the unobserved nodes of the network. When the condition number is sufficiently large, reconstructing the trajectory from observations from the subset will be infeasible. This knowledge can be used to choose an optimal subset from which to observe a network.

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.

A symbolic network-based nonlinear theory for dynamical systems observability

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 observability becomes a computationally prohibitive task. Our approach consists in computing the observability coefficients from a symbolic Jacobian matrix whose elements encode the linear, nonlinear polynomial or rational nature of the interaction among the variables. The novelty we introduce in this paper, required for treating large-dimensional systems, is to identify from the symbolic Jacobian matrix the minimal set of variables (together with their time derivatives) candidate to be measured for completing the state space reconstruction. Then symbolic observability coefficients are computed from the symbolic observability matrix. Our results are in agreement with the analytical computations, evidencing the correctness of our approach. Its application to efficiently exploring the dynamics of real world complex systems such as power grids, socioeconomic networks or biological networks is quite promising.

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.

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.