  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 the 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.
