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On (Eventually) Monotone Dynamical Systems and Positive Koopman Semigroups Sootla, Aivar ; Mauroy, Alexandre Conference (2016, July) Monotone systems are dynamical systems whose solutions preserve a partial order in initial conditions for all times. It stands to reason that some systems may preserve a partial order only after an ... [more ▼] Monotone systems are dynamical systems whose solutions preserve a partial order in initial conditions for all times. It stands to reason that some systems may preserve a partial order only after an initial transient. These systems are usually called eventually monotone. While monotone systems have an easy characterization in terms of the sign pattern of the Jacobian matrix (i.e. Kamke-M\"uller condition), eventually monotone systems have not been characterized in such an explicit manner. In order to provide such 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, a spectral characterization of nonlinear eventually monotone systems is not straightforward, but can be obtained in the framework of the so-called Koopman operator. Additionally, we explore connections between (eventual) monotonicity and (eventual) positivity of the Koopman semigroup. This allows to view our results as a generalization of the Perron-Frobenius theory to nonlinear dynamical systems. We consider a biologically inspired example to illustrate the applicability of eventual monotonicity. [less ▲] Detailed reference viewed: 11 (3 ULg)Global stability analysis for nonlinear systems using the eigenfunctions of the Koopman operator / Neuroscience applications: isochrons and isostables Mauroy, Alexandre Conference (2015, July) Detailed reference viewed: 47 (3 ULg)Computation of the Koopman eigenfunctions is a systematic method for global stability analysis Mauroy, Alexandre ; Conference (2015, May) Detailed reference viewed: 58 (2 ULg)Extreme phase sensitivity in systems with fractal isochrons Mauroy, Alexandre ; in Physica D: Nonlinear Phenomena (2015), 308 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 17 (0 ULg)Converging to and escaping from the global equilibrium: Isostables and optimal control Mauroy, Alexandre in Proceedings of the 53rd IEEE Conference on Decision and Control (2014, December) This paper studies the optimal control of trajectories converging to or escaping from a stable equilibrium. The control duration is assumed to be short. When the control is turned off, the trajectories ... [more ▼] This paper studies the optimal control of trajectories converging to or escaping from a stable equilibrium. The control duration is assumed to be short. When the control is turned off, the trajectories have not reached the target and they subsequently evolve according to the free motion dynamics. In this context, we show that the problem can be formulated as a finite-horizon optimal control problem which relies on the notion of isostables. For linear and nonlinear systems, we solve this problem using Pontryagin’s maximum principle and we study the relationship between the optimal solutions and the geometry of the isostables. Finally, optimal strategies for choosing the magnitude and duration of the control are considered. [less ▲] Detailed reference viewed: 8 (0 ULg)Global Isochrons and Phase Sensitivity of Bursting Neurons Mauroy, Alexandre ; ; et al in SIAM Journal on Applied Dynamical Systems (2014), 13(1), 306-338 Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models. One popular way of capturing phase sensitivity is through the computation of isochrons ... [more ▼] Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models. One popular way of capturing phase sensitivity is through the computation of isochrons---subsets of the state space that each converge to the same trajectory on the limit cycle. However, the computation of isochrons is notoriously difficult, especially for bursting neuron models. In [W. E. Sherwood and J. Guckenheimer, SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 659--703], the phase sensitivity of the bursting Hindmarsh--Rose model is studied through the use of singular perturbation theory: cross sections of the isochrons of the full system are approximated by those of fast subsystems. In this paper, we complement the previous study, providing a detailed phase sensitivity analysis of the full (three-dimensional) system, including computations of the full (two-dimensional) isochrons. To our knowledge, this is the first such computation for a bursting neuron model. This was made possible thanks to the numerical method recently proposed in [A. Mauroy and I. Mezić, Chaos, 22 (2012), 033112]---relying on the spectral properties of the so-called Koopman operator---which is complemented with the use of adaptive quadtree and octree grids. The main result of the paper is to highlight the existence of a region of high phase sensitivity called the almost phaseless set and to completely characterize its geometry. In particular, our study reveals the existence of a subset of the almost phaseless set that is not predicted by singular perturbation theory (i.e., by the isochrons of fast subsystems). We also discuss how the almost phaseless set is related to empirically observed phenomena such as addition/deletion of spikes and to extrema of the phase response of the system. Finally, through the same numerical method, we show that an elliptic bursting model is characterized by a very high phase sensitivity and other remarkable properties. [less ▲] Detailed reference viewed: 20 (0 ULg)Spectral operator-theoretic description of nonlinear systems: a systematic approach to global stability analysis Mauroy, Alexandre ; Poster (2014) Detailed reference viewed: 13 (0 ULg)A spectral operator-theoretic framework for global stability Mauroy, Alexandre ; in Proceedings of the IEEE Conference on Decision and Control (2013, December) The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has ... [more ▼] The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples. [less ▲] Detailed reference viewed: 11 (2 ULg)Isochrons and isostables of dynamical systems: Relationship to Koopman operator spectrum, Mauroy, Alexandre ; Conference (2013) Detailed reference viewed: 12 (0 ULg)Global analysis of a continuum model for monotone pulse-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe in IEEE Transactions on Automatic Control (2013), 58(5), 1154-1166 We consider a continuum of phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulse-coupledmodels, the stability results ... [more ▼] We consider a continuum of phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulse-coupledmodels, 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 show 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. © 2012 IEEE. [less ▲] Detailed reference viewed: 32 (1 ULg)Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics Mauroy, Alexandre ; ; in Physica D: Nonlinear Phenomena (2013), 261 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 ... [more ▼] 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 the 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. © 2013 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 14 (2 ULg)Kick synchronization versus diffusive synchronization Mauroy, Alexandre ; Sacré, Pierre ; Sepulchre, Rodolphe in Proceedings of the 51st IEEE Conference on Decision and Control (invited tutorial session) (2012, December) The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on ... [more ▼] The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on synchronization, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist. The paper stresses fundamental differences between the two models, such as the different contraction measures underlying the analysis, as well as important analogies that can be drawn in the limit of weak coupling. [less ▲] Detailed reference viewed: 36 (5 ULg)Contraction of monotone phase-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe in Systems & Control Letters (2012), 61(11), 1097-1102 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 ... [more ▼] 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. © 2012 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 10 (0 ULg)On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics Mauroy, Alexandre ; in Chaos (2012), 22(3), 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 ... [more ▼] 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. © 2012 American Institute of Physics. [less ▲] Detailed reference viewed: 12 (0 ULg)On the dichotomic collective behaviors of large populations of pulse-coupled firing oscillators Mauroy, Alexandre Doctoral thesis (2011) The study of populations of pulse-coupled firing oscillators is a general and simple paradigm to investigate a wealth of natural phenomena, including the collective behaviors of neurons, the ... [more ▼] The study of populations of pulse-coupled firing oscillators is a general and simple paradigm to investigate a wealth of natural phenomena, including the collective behaviors of neurons, the synchronization of cardiac pacemaker cells, or the dynamics of earthquakes. In this framework, the oscillators of the network interact through an instantaneous impulsive coupling: whenever an oscillator fires, it sends out a pulse which instantaneously increments the state of the other oscillators by a constant value. There is an extensive literature on the subject, which investigates various model extensions, but only in the case of leaky integrate-and-fire oscillators. In contrast, the present dissertation addresses the study of other integrate-and-fire dynamics: general monotone integrate-and-fire dynamics and quadratic integrate-and-fire dynamics. The main contribution of the thesis highlights that the populations of oscillators exhibit a dichotomic collective behavior: either the oscillators achieve perfect synchrony (slow firing frequency) or the oscillators converge toward a phase-locked clustering configuration (fast firing frequency). The dichotomic behavior is established both for finite and infinite populations of oscillators, drawing a strong parallel between discrete-time systems in finite-dimensional spaces and continuous-time systems in infinite-dimensional spaces. The first part of the dissertation is dedicated to the study of monotone integrate-and-fire dynamics. We show that the dichotomic behavior of the oscillators results from the monotonicity property of the dynamics: the monotonicity property induces a global contraction property of the network, that forces the dichotomic behavior. Interestingly, the analysis emphasizes that the contraction property is captured through a 1-norm, instead of a (more common) quadratic norm. In the second part of the dissertation, we investigate the collective behavior of quadratic integrate-and-fire oscillators. Although the dynamics is not monotone, an “average” monotonicity property ensures that the collective behavior is still dichotomic. However, a global analysis of the dichotomic behavior is elusive and leads to a standing conjecture. A local stability analysis circumvents this issue and proves the dichotomic behavior in particular situations (small networks, weak coupling, etc.). Surprisingly, the local stability analysis shows that specific integrate-and-fire oscillators exhibit a non-dichotomic behavior, thereby suggesting that the dichotomic behavior is not a general feature of every network of pulse-coupled oscillators. The present thesis investigates the remarkable dichotomic behavior that emerges from networks of pulse-coupled integrate-and-fire oscillators, putting emphasis on the stability properties of these particular networks and developing theoretical results for the analysis of the corresponding dynamical systems. [less ▲] Detailed reference viewed: 146 (22 ULg)Existence and stability of a dichotomic behavior in infinite populations of pulse-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe Poster (2011) Detailed reference viewed: 4 (0 ULg)A transport equation for pulse-coupled phase oscillators and a Lyapunov function for its global analysis Mauroy, Alexandre ; Sepulchre, Rodolphe Conference (2011) Detailed reference viewed: 2 (0 ULg)Existence and stability of a dichotomic behavior in infinite populations of pulse-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe Conference (2011) Detailed reference viewed: 3 (0 ULg)Local stability results for the collective behaviors of infinite populations of pulse-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe in Proceedings of the IEEE Conference on Decision and Control (2011) In this paper, we investigate the behavior of pulse-coupled integrate-and-fire oscillators. Because the stability analysis of finite populations is intricate, we investigate stability results in the ... [more ▼] In this paper, we investigate the behavior of pulse-coupled integrate-and-fire oscillators. Because the stability analysis of finite populations is intricate, we investigate stability results in the approximation of infinite populations. In addition to recovering known stability results of finite populations, we also obtain new stability results for infinite populations. In particular, under a weak coupling assumption, we solve for the continuum model a conjecture still prevailing in the finite dimensional case. © 2011 IEEE. [less ▲] Detailed reference viewed: 6 (0 ULg)Global Analysis of Firing Maps Mauroy, Alexandre ; ; et al in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (2010, July) In this paper, we study the behavior of pulse-coupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper ... [more ▼] In this paper, we study the behavior of pulse-coupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper threshold, it is reset to the lower threshold and emits a pulse which increments by a constant value the state of every other oscillator. The behavior of the system is described by the so-called firing map: depending on the stability of the firing map, an important dichotomy characterizes the behavior of the oscillators (synchronization or clustering). The firing map is the composition of a linear map with a scalar nonlinearity. After briefly discussing the case of the scalar firing map (corresponding to two oscillators), the stability analysis is extended to the general n-dimensional firing map (for n +1 oscillators). Different models are considered (leaky oscillators, quadratic oscillators,...), with a particular emphasis on the persistence of the dichotomy in higher dimensions. [less ▲] Detailed reference viewed: 74 (18 ULg) |
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