On the use of Fourier averages to compute the global isochrons
of (quasi)periodic dynamics
A. Mauroy
a)
and I. Mezic´
b)
Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara,
California 93106, USA
(Received 27 March 2012; accepted 25 June 2012; published online 18 July 2012)
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.
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2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4736859]
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 iso-
chrons. The computation of isochrons is particularly intri-
cate 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 associ-
ated with the system. We apply the method to various
examples and also extend the framework to quasiperiodic
systems.
I. INTRODUCTION
According to the seminal works,
8,23
a dynamical system
with a stable limit cycle can be reduced to a phase model. In
other words, a limit-cycle oscillator (evolving in a high-
dimensional space) is equivalent to a phase oscillator (evolv-
ing on the one-dimensional circle), a reduced model that is
more amenable to mathematical analysis (see Ref. 5 for a
review). Phase reduction thereby appears as a useful frame-
work to analyze the sensitivity and the robustness of limit
cycles,
20
to compare and design models of oscillators,
14
and
to study the collective behaviors of interacting oscillators
(see, e.g., the Kuramoto model
17
).
A phase model is obtained by assigning a phase variable
to each point in the basin of attraction of the limit cycle.
First, the phase is defined on the limit cycle (up to a given
reference) as the quantity proportional to the time spent on
the cycle. Then, the notion of phase is extended to the whole
basin of attraction through the isochrons (the term was orig-
inally coined in Ref. 24), a concept that corresponds to the
invariant fibration of the stable manifold of the limit cycle.
In other words, the isochrons partition the basin of attraction
in such a way that trajectories starting from the same iso-
chron asymptotically converge to the same orbit on the limit
cycle.
The computation of isochrons is desirable not only to
obtain a phase reduction of the system but also to provide a
global picture of the system dynamics as well as an insight
into the sensitivity to external perturbations. Local iso-
chrons—in the vicinity of the limit cycle—are obtained using
either linearization techniques (and higher order approxima-
tions
18,19
) or standard backward integration methods (see Ref.
6 for a detailed example of the algorithm). In contrast, the
computation of global isochrons—in the entire basin of attrac-
tion—is much more involved and has only been investigated
in a few studies, most of which propose numerical schemes
based on the extension of local isochrons through backward
integration.
2,16
In addition, the computation of global iso-
chrons is particularly difficult for dynamics with multiple
time scales (slow-fast dynamics) and for high-dimensional
dynamics, two situations where backward integration is ineffi-
cient owing to numerical sensitivity issues. While an elegant
method is provided in Ref. 12 to compute the global isochrons
a)
Electronic mail: alex.mauroy@engr.ucsb.edu.
b)
Electronic mail: mezic@engr.ucsb.edu.
1054-1500/2012/22(3)/033112/9/$30.00
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2012 American Institute of Physics22, 033112-1
CHAOS 22, 033112 (2012)