On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics

Abstract

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.

Global Isochrons and Phase Sensitivity of Bursting Neurons

“…The idea of computing the phase function Θ and isochrons of a spatially complex dynamical system via the method of Fourier averages was first proposed in [31]. In this framework, the function Θ is related to an eigenfunction of the so-called Koopman operator [32].…”

“…The results are obtained with the numerical method recently proposed in [31], which computes the isochrons as the level sets of Fourier averages evaluated along the system trajectories, utilizing the Koopman operator framework for dynamical systems analysis [4,32]. This method is complemented with the use of adaptive grids (quadtree-and octree-based), which enable better detail in regions where the isochrons are more dense.…”

A spectral operator-theoretic framework for global stability

“…According to (8), they satisfy V(ϕ t (x)) ≤ exp(ℜ{λ * 1 }t)V(x) for x ∈ X, where λ * 1 is the eigenvalue closest to the imaginary axis. Note that these Lyapunov functions are not necessarily smooth.…”

“…The other eigenfunctions, associated with purely imaginary eigenvalues, provide no information on stability but are related to asymptotic properties of the trajectories on the attractor. For instance, they can be used to compute periodic invariant sets on the attractor [10] or to compute the so-called isochrons of (quasi)periodic attractors [8].…”

“…However, while these properties were successfully used to capture the system behavior on the attractor (see e.g. [8], [10]), they have never been exploited in the context of stability theory.…”