Spectral identification of networks using sparse measurements

We propose a new method to recover global information about a network of interconnected dynamical systems based on observations made at a small number (possibly one) of its nodes. In contrast to classical identification of full graph topology, we focus on the identification of the spectral graph-theoretic properties of the network, a framework that we call spectral network identification. The main theoretical results connect the spectral properties of the network to the spectral properties of the dynamics, which are well-defined in the context of the so-called Koopman operator and can be extracted from data through the Dynamic Mode Decomposition algorithm. These results are obtained for networks of diffusively-coupled units that admit a stable equilibrium state. For large networks, a statistical approach is considered, which focuses on spectral moments of the network and is well-suited to the case of heterogeneous populations. Our framework provides efficient numerical methods to infer global information on the network from sparse local measurements at a few nodes. Numerical simulations show for instance the possibility of detecting the mean number of connections or the addition of a new vertex using measurements made at one single node, that need not be representative of the other nodes' properties.Comment: 3

Global computation of phase-amplitude reduction for limit-cycle dynamics

Recent years have witnessed increasing interest to phase-amplitude reduction of limit-cycle dynamics. Adding an amplitude coordinate to the phase coordinate allows to take into account the dynamics transversal to the limit cycle and thereby overcomes the main limitations of classic phase reduction (strong convergence to the limit cycle and weak inputs). While previous studies mostly focus on local quantities such as infinitesimal responses, a major and limiting challenge of phase-amplitude reduction is to compute amplitude coordinates globally, in the basin of attraction of the limit cycle. In this paper, we propose a method to compute the full set of phase-amplitude coordinates in the large. Our method is based on the so-called Koopman (composition) operator and aims at computing the eigenfunctions of the operator through Laplace averages (in combination with the harmonic balance method). This yields a forward integration method that is not limited to two-dimensional systems. We illustrate the method by computing the so-called isostables of limit cycles in two, three, and four-dimensional state spaces, as well as their responses to strong external inputs.Comment: 26 page

Koopman-based lifting techniques for nonlinear systems identification

We develop a novel lifting technique for nonlinear system identification based on the framework of the Koopman operator. The key idea is to identify the linear (infinite-dimensional) Koopman operator in the lifted space of observables, instead of identifying the nonlinear system in the state space, a process which results in a linear method for nonlinear systems identification. The proposed lifting technique is an indirect method that does not require to compute time derivatives and is therefore well-suited to low-sampling rate datasets. Considering different finite-dimensional subspaces to approximate and identify the Koopman operator, we propose two numerical schemes: a main method and a dual method. The main method is a parametric identification technique that can accurately reconstruct the vector field of a broad class of systems. The dual method provides estimates of the vector field at the data points and is well-suited to identify high-dimensional systems with small datasets. The present paper describes the two methods, provides theoretical convergence results, and illustrates the lifting techniques with several examples

An operator-theoretic approach to differential positivity

Differentially positive systems are systems whose linearization along trajectories is positive. Under mild assumptions, their solutions asymptotically converge to a one-dimensional attractor, which must be a limit cycle in the absence of fixed points in the limit set. In this paper, we investigate the general connections between the (geometric) properties of differentially positive systems and the (spectral) properties of the Koopman operator. In particular, we obtain converse results for differential positivity, showing for instance that any hyperbolic limit cycle is differentially positive in its basin of attraction. We also provide the construction of a contracting cone field.A. Mauroy holds a BELSPO Return Grant and F. Forni holds a FNRS fellowship. This paper presents research results of the Belgian Network DYSCO, funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/CDC.2015.740332

Contraction of monotone phase-coupled oscillators

This paper establishes a global contraction property for networks of phase-coupled oscillators characterized by a monotone coupling function. The contraction measure is a total variation distance. The contraction property determines the asymptotic behavior of the network, which is either finite-time synchronization or asymptotic convergence to a splay state.Comment: 10 page

A spectral characterization of nonlinear normal modes

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity.Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy OfficeThis is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jsv.2016.05.01

Terminology - glossary including acronyms and quotations in use for the conservative spinal deformities treatment: 8th SOSORT consensus paper

<p>Abstract</p> <p>Background</p> <p>This report is the SOSORT Consensus Paper on Terminology for use in the treatment of conservative spinal deformities. Figures are provided and relevant literature is cited where appropriate.</p> <p>Methods</p> <p>The Delphi method was used to reach a preliminary consensus before the meeting, where the terms that still needed further clarification were discussed.</p> <p>Results</p> <p>A final agreement was found for all the terms, which now constitute the base of this glossary. New terms will be added after being discussed and accepted.</p> <p>Discussion</p> <p>When only one set of terms is used for communication in a place or among a group of people, then everyone can clearly and efficiently communicate. This principle applies for any professional group. Until now, no common set of terms was available in the field of the conservative treatment of scoliosis and spinal deformities. This glossary gives a common base language to draw from to discuss data, findings and treatment.</p

The origin of the allometric scaling of lung ventilation in mammals

A model of optimal control of ventilation recently developed for humans has suggested that the localization of the transition between a convective and a diffusive transport of the respiratory gas determines how ventilation should be controlled to minimize its energetic cost at any metabolic regime. We generalized this model to any mammal, based on the core morphometric characteristics shared by all mammals' lungs and on their allometric scaling from the literature. Since the main energetic costs of ventilation are related to the convective transport, we prove that, for all mammals, the localization of the shift from a convective transport into a diffusive transport plays a critical role on keeping that cost low while fulfilling the lung function. Our model predicts for the first time where this transition zone should occur in order to minimize the energetic cost of ventilation, depending on the mammals' mass and on the metabolic regime. From that optimal localization, we are able to derive predicted allometric scaling laws for both tidal volumes and breathing rates, at any metabolic regime. We ran our model for the three common metabolic rates -- basal, field and maximal -- and showed that our predictions accurately reproduce the experimental data available in the literature. Our analysis supports the hypothesis that the mammals' allometric scaling laws of tidal volumes and breathing rates at a given metabolic rate are driven by a few core geometrical characteristics shared by the mammals' lungs and the physical processes of the respiratory gas transport

Interplay between geometry and flow distribution in an airway tree

Uniform fluid flow distribution in a symmetric volume can be realized through a symmetric branched tree. It is shown here, however, that the flow partitioning can be highly sensitive to deviations from exact symmetry if inertial effects are present. This is found by direct numerical simulation of the Navier-Stokes equations in a 3D tree geometry. The flow asymmetry is quantified and found to depend on the Reynolds number. Moreover, for a given Reynolds number, we show that the flow distribution depends on the aspect ratio of the branching elements as well as their angular arrangement. Our results indicate that physiological variability should be severely restricted in order to ensure uniform fluid distribution in a tree. This study suggests that any non-uniformity in the air flow distribution in human lungs should be influenced by the respiratory conditions, rest or hard exercise