80 research outputs found
Spectral identification of networks with inputs
We consider a network of interconnected dynamical systems. Spectral network
identification consists in recovering the eigenvalues of the network Laplacian
from the measurements of a very limited number (possibly one) of signals. These
eigenvalues allow to deduce some global properties of the network, such as
bounds on the node degree.
Having recently introduced this approach for autonomous networks of nonlinear
systems, we extend it here to treat networked systems with external inputs on
the nodes, in the case of linear dynamics. This is more natural in several
applications, and removes the need to sometimes use several independent
trajectories. We illustrate our framework with several examples, where we
estimate the mean, minimum, and maximum node degree in the network. Inferring
some information on the leading Laplacian eigenvectors, we also use our
framework in the context of network clustering.Comment: 8 page
Extreme phase sensitivity in systems with fractal isochrons
Sensitivity to initial conditions is usually associated with chaotic dynamics
and strange attractors. However, even systems with (quasi)periodic dynamics can
exhibit it. In this context we report on the fractal properties of the
isochrons of some continuous-time asymptotically periodic systems. We define a
global measure of phase sensitivity that we call the phase sensitivity
coefficient and show that it is an invariant of the system related to the
capacity dimension of the isochrons. Similar results are also obtained with
discrete-time systems. As an illustration of the framework, we compute the
phase sensitivity coefficient for popular models of bursting neurons,
suggesting that some elliptic bursting neurons are characterized by isochrons
of high fractal dimensions and exhibit a very sensitive (unreliable) phase
response.Comment: 32 page
Global analysis of a continuum model for monotone pulse-coupled oscillators
We consider a continuum of phase oscillators on the circle interacting
through an impulsive instantaneous coupling. In contrast with previous studies
on related pulse-coupled models, the stability results obtained in the
continuum limit are global. For the nonlinear transport equation governing the
evolution of the oscillators, we propose (under technical assumptions) a global
Lyapunov function which is induced by a total variation distance between
quantile densities. The monotone time evolution of the Lyapunov function
completely characterizes the dichotomic behavior of the oscillators: either the
oscillators converge in finite time to a synchronous state or they
asymptotically converge to an asynchronous state uniformly spread on the
circle. The results of the present paper apply to popular phase oscillators
models (e.g. the well-known leaky integrate-and-fire model) and draw a strong
parallel between the analysis of finite and infinite populations. In addition,
they provide a novel approach for the (global) analysis of pulse-coupled
oscillators.Comment: 33 page
Linear identification of nonlinear systems: A lifting technique based on the Koopman operator
We exploit the key idea that nonlinear system identification is equivalent to
linear identification of the socalled Koopman operator. Instead of considering
nonlinear system identification in the state space, we obtain a novel linear
identification technique by recasting the problem in the infinite-dimensional
space of observables. This technique can be described in two main steps. In the
first step, similar to the socalled Extended Dynamic Mode Decomposition
algorithm, the data are lifted to the infinite-dimensional space and used for
linear identification of the Koopman operator. In the second step, the obtained
Koopman operator is "projected back" to the finite-dimensional state space, and
identified to the nonlinear vector field through a linear least squares
problem. The proposed technique is efficient to recover (polynomial) vector
fields of different classes of systems, including unstable, chaotic, and open
systems. In addition, it is robust to noise, well-suited to model low sampling
rate datasets, and able to infer network topology and dynamics.Comment: 6 page
Geometric Properties of Isostables and Basins of Attraction of Monotone Systems
In this paper, we study geometric properties of basins of attraction of
monotone systems. Our results are based on a combination of monotone systems
theory and spectral operator theory. We exploit the framework of the Koopman
operator, which provides a linear infinite-dimensional description of nonlinear
dynamical systems and spectral operator-theoretic notions such as eigenvalues
and eigenfunctions. The sublevel sets of the dominant eigenfunction form a
family of nested forward-invariant sets and the basin of attraction is the
largest of these sets. The boundaries of these sets, called isostables, allow
studying temporal properties of the system. Our first observation is that the
dominant eigenfunction is increasing in every variable in the case of monotone
systems. This is a strong geometric property which simplifies the computation
of isostables. We also show how variations in basins of attraction can be
bounded under parametric uncertainty in the vector field of monotone systems.
Finally, we study the properties of the parameter set for which a monotone
system is multistable. Our results are illustrated on several systems of two to
four dimensions.Comment: 12 pages, to appear in IEEE Transaction on Automatic Contro
Operator-Theoretic Characterization of Eventually Monotone Systems
Monotone systems are dynamical systems whose solutions preserve a partial
order in the initial condition for all positive times. It stands to reason that
some systems may preserve a partial order only after some initial transient.
These systems are usually called eventually monotone. While monotone systems
have a characterization in terms of their vector fields (i.e. Kamke-Muller
condition), eventually monotone systems have not been characterized in such an
explicit manner. In order to provide a characterization, we drew inspiration
from the results for linear systems, where eventually monotone (positive)
systems are studied using the spectral properties of the system (i.e.
Perron-Frobenius property). In the case of nonlinear systems, this spectral
characterization is not straightforward, a fact that explains why the class of
eventually monotone systems has received little attention to date. In this
paper, we show that a spectral characterization of nonlinear eventually
monotone systems can be obtained through the Koopman operator framework. We
consider a number of biologically inspired examples to illustrate the potential
applicability of eventual monotonicity.Comment: 13 page
Shaping Pulses to Control Bistable Monotone Systems Using Koopman Operator
In this paper, we further develop a recently proposed control method to
switch a bistable system between its steady states using temporal pulses. The
motivation for using pulses comes from biomedical and biological applications
(e.g. synthetic biology), where it is generally difficult to build feedback
control systems due to technical limitations in sensing and actuation. The
original framework was derived for monotone systems and all the extensions
relied on monotone systems theory. In contrast, we introduce the concept of
switching function which is related to eigenfunctions of the so-called Koopman
operator subject to a fixed control pulse. Using the level sets of the
switching function we can (i) compute the set of all pulses that drive the
system toward the steady state in a synchronous way and (ii) estimate the time
needed by the flow to reach an epsilon neighborhood of the target steady state.
Additionally, we show that for monotone systems the switching function is also
monotone in some sense, a property that can yield efficient algorithms to
compute it. This observation recovers and further extends the results of the
original framework, which we illustrate on numerical examples inspired by
biological applications.Comment: 7 page
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