For asymptotically periodic systems, a powerful (phase) reduction of the dynamics is obtained by computing the so-called isochrons, i.e. the sets of points that converge toward the same trajectory on the limit cycle. Motivated by the analysis of excitable systems, a similar reduction has been attempted for non-periodic systems admitting a stable fixed point. In this case, the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories. Instead, the sets of interest-that we call "isostables"-are defined in literature as the sets of points that converge toward the same trajectory on a stable slow manifold of the fixed point. However, it turns out that this definition of the isostables holds only for systems with slow-fast dynamics.Also, efficient methods for computing the isostables are missing.The present paper provides a general framework for the definition and the computation of the isostables of stable fixed points, which is based on the spectral properties of the so-called Koopman operator. More precisely, the isostables are defined as the level sets of a particular eigenfunction of the Koopman operator. Through this approach, the isostables are unique and well-defined objects related to the asymptotic properties of the system. Also, the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action-angle coordinates for the dynamics. In addition, an efficient algorithm for computing the isostables is obtained, which relies on the evaluation of Laplace averages along the trajectories. The method is illustrated with the excitable FitzHugh-Nagumo model and with the Lorenz model. Finally, we discuss how these methods based on the Koopman operator framework relate to the global linearization of the system and to the derivation of special Lyapunov functions.
We propose a novel operator-theoretic framework to study global stability of nonlinear systems.Based on the spectral properties of the so-called Koopman operator, our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems. The main results establish the (necessary and sufficient) relationship between the existence of specific eigenfunctions of the Koopman operator and the global stability property of hyperbolic fixed points and limit cycles. These results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space.A. Mauroy is with the
The concept of isochrons is crucial for the analysis of asymptotically periodic systems. Roughly, isochrons are sets of points that partition the basin of attraction of a limit cycle according to the asymptotic behavior of the trajectories. The computation of global isochrons (in the whole basin of attraction) is however difficult, and the existing methods are inefficient in high-dimensional spaces. In this context, we present a novel (forward integration) algorithm for computing the global isochrons of high-dimensional dynamics, which is based on the notion of Fourier time averages evaluated along the trajectories. Such Fourier averages in fact produce eigenfunctions of the Koopman semigroup associated with the system, and isochrons are obtained as level sets of those eigenfunctions. The method is supported by theoretical results and validated by several examples of increasing complexity, including the 4-dimensional Hodgkin-Huxley model. In addition, the framework is naturally extended to the study of quasiperiodic systems and motivates the definition of generalized isochrons of the torus. This situation is illustrated in the case of two coupled Van der Pol oscillators. An efficient way to study asymptotically periodic systems is to consider phase differences between the trajectories, a framework that leads to a powerful dimensional reduction of the system to a one-dimensional model. However, the price to pay for such a reduction is the computation of particular sets of the state space, i.e., the so-called isochrons. The computation of isochrons is particularly intricate in high-dimensional spaces-where the isochrons can exhibit a complex geometry-and the existing methods are typically limited to 2-dimensional systems. In contrast, this paper proposes a novel algorithm that is well-suited to the computation of isochrons in high-dimensional spaces. More precisely, we show that the isochrons can be obtained through the computation of Fourier averages evaluated along the trajectories. As a consequence, we obtain a relationship between isochrons and level sets of a class of eigenfunctions of the Koopman semigroup associated with the system. We apply the method to various examples and also extend the framework to quasiperiodic systems.
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 a component of the 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.