173 research outputs found
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
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
Shape Self-Regulation in Early Lung Morphogenesis
The arborescent architecture of mammalian conductive airways results from the repeated branching of lung endoderm into surrounding mesoderm. Subsequent lung’s striking geometrical features have long raised the question of developmental mechanisms involved in morphogenesis. Many molecular actors have been identified, and several studies demonstrated the central role of Fgf10 and Shh in growth and branching. However, the actual branching mechanism and the way branching events are organized at the organ scale to achieve a self-avoiding tree remain to be understood through a model compatible with evidenced signaling. In this paper we show that the mere diffusion of FGF10 from distal mesenchyme involves differential epithelial proliferation that spontaneously leads to branching. Modeling FGF10 diffusion from sub-mesothelial mesenchyme where Fgf10 is known to be expressed and computing epithelial and mesenchymal growth in a coupled manner, we found that the resulting laplacian dynamics precisely accounts for the patterning of FGF10-induced genes, and that it spontaneously involves differential proliferation leading to a self-avoiding and space-filling tree, through mechanisms that we detail. The tree’s fine morphological features depend on the epithelial growth response to FGF10, underlain by the lung’s complex regulatory network. Notably, our results suggest that no branching information has to be encoded and that no master routine is required to organize branching events at the organ scale. Despite its simplicity, this model identifies key mechanisms of lung development, from branching to organ-scale organization, and could prove relevant to the development of other branched organs relying on similar pathways
Chemical Power for Microscopic Robots in Capillaries
The power available to microscopic robots (nanorobots) that oxidize
bloodstream glucose while aggregated in circumferential rings on capillary
walls is evaluated with a numerical model using axial symmetry and
time-averaged release of oxygen from passing red blood cells. Robots about one
micron in size can produce up to several tens of picowatts, in steady-state, if
they fully use oxygen reaching their surface from the blood plasma. Robots with
pumps and tanks for onboard oxygen storage could collect oxygen to support
burst power demands two to three orders of magnitude larger. We evaluate
effects of oxygen depletion and local heating on surrounding tissue. These
results give the power constraints when robots rely entirely on ambient
available oxygen and identify aspects of the robot design significantly
affecting available power. More generally, our numerical model provides an
approach to evaluating robot design choices for nanomedicine treatments in and
near capillaries.Comment: 28 pages, 7 figure
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