14
H. Nakao
Recent developments in the Koopman operator approach to nonlinear dynamical
systems [18–20] have shown that the above definition is actually natural in the sense
that R is given by an eigenfunction of the infinitesimal Koopman operator F(X) · ∇ X
associated with the eigenvalue λ, and such R has been calculated e.g. for the van
der Pol oscillator. The level sets of R are called isostables, in a similar sense to
the isochrons for the asymptotic phase. In general, we have n − 1 amplitudes, or
principal Koopman eigenfunctions, associated with Floquet exponents λ 2 , . . . , λ n ,
which can take complex values. Accordingly, the range of the amplitudes should
be taken as C rather than R. It is easy to show that the exponential of the phase
function, = e
i , is also an eigenfunction of F(X) · ∇ X with eigenvalue iω. Thus,
the Koopman operator approach gives a unifying viewpoint on the phase-amplitude
description, or global linearization, of the flows around stable limit cycles.
Having defined the phase θ = (X) and the amplitude r = R(X) for X ∈ B,
we can derive approximate phase and amplitude equations for a weakly perturbed
limit-cycle oscillator described by
˙
X(t) = F(X(t)) + p(X(t), t),
(2.4)
where p ∈ R
n represents the perturbation applied to the oscillator and > 0 is a
small parameter. The equation for the phase can be expressed as ˙
θ = ∇ X (X) ·
˙
X = ∇ X (X) · F(X) + ∇ X (X) · p = ω + ∇ X (X) · p, which still depends on
X. To obtain an equation closed in θ, we use the fact that X is near X 0 (θ) on χ and
approximate X as X = X 0 (θ) + O(). Then, up to the lowest-order O() in , the
phase θ obeys
˙
θ(t) ω + Z(θ(t)) · p(X 0 (θ(t)), t),
(2.5)
where we defined the phase sensitivity function Z(θ) = ∇ X (X)| X=X 0 (θ) . Likewise, the amplitude obeys ˙
r = ∇ X R(X) · ˙
X = ∇ X R(X) · F(X) + ∇ X R(X) · p =
λr + ∇ X R(X) · p and, by assuming that X is in the neighborhood of X 0 (θ), we
obtain
˙
r (t) λr (t) + I(θ(t)) · p(X 0 (θ(t)), t),
(2.6)
which is again correct up to O(). We here defined I(θ) = ∇ X R(X)| X=X 0 (θ) , which
we call the amplitude sensitivity function.
It is difficult to fully determine the phase function (X) and the amplitude function
R(X) for all X ∈ B even numerically in multidimensional systems. However, if we
are interested in the weakly perturbed case, Eq. (2.4), we only need the functions Z
and I. It can be shown that these quantities are given by 2π-periodic solutions to the
following adjoint linear ODEs [3, 5, 16, 24, 30, 38]:
ω
d
dθ
Z(θ) = −J (θ)
Z(θ), ω
d
dθ
I(θ) = −[J (θ)
− λ]I(θ),
(2.7)
H. Nakao
Recent developments in the Koopman operator approach to nonlinear dynamical
systems [18–20] have shown that the above definition is actually natural in the sense
that R is given by an eigenfunction of the infinitesimal Koopman operator F(X) · ∇ X
associated with the eigenvalue λ, and such R has been calculated e.g. for the van
der Pol oscillator. The level sets of R are called isostables, in a similar sense to
the isochrons for the asymptotic phase. In general, we have n − 1 amplitudes, or
principal Koopman eigenfunctions, associated with Floquet exponents λ 2 , . . . , λ n ,
which can take complex values. Accordingly, the range of the amplitudes should
be taken as C rather than R. It is easy to show that the exponential of the phase
function, = e
i , is also an eigenfunction of F(X) · ∇ X with eigenvalue iω. Thus,
the Koopman operator approach gives a unifying viewpoint on the phase-amplitude
description, or global linearization, of the flows around stable limit cycles.
Having defined the phase θ = (X) and the amplitude r = R(X) for X ∈ B,
we can derive approximate phase and amplitude equations for a weakly perturbed
limit-cycle oscillator described by
˙
X(t) = F(X(t)) + p(X(t), t),
(2.4)
where p ∈ R
n represents the perturbation applied to the oscillator and > 0 is a
small parameter. The equation for the phase can be expressed as ˙
θ = ∇ X (X) ·
˙
X = ∇ X (X) · F(X) + ∇ X (X) · p = ω + ∇ X (X) · p, which still depends on
X. To obtain an equation closed in θ, we use the fact that X is near X 0 (θ) on χ and
approximate X as X = X 0 (θ) + O(). Then, up to the lowest-order O() in , the
phase θ obeys
˙
θ(t) ω + Z(θ(t)) · p(X 0 (θ(t)), t),
(2.5)
where we defined the phase sensitivity function Z(θ) = ∇ X (X)| X=X 0 (θ) . Likewise, the amplitude obeys ˙
r = ∇ X R(X) · ˙
X = ∇ X R(X) · F(X) + ∇ X R(X) · p =
λr + ∇ X R(X) · p and, by assuming that X is in the neighborhood of X 0 (θ), we
obtain
˙
r (t) λr (t) + I(θ(t)) · p(X 0 (θ(t)), t),
(2.6)
which is again correct up to O(). We here defined I(θ) = ∇ X R(X)| X=X 0 (θ) , which
we call the amplitude sensitivity function.
It is difficult to fully determine the phase function (X) and the amplitude function
R(X) for all X ∈ B even numerically in multidimensional systems. However, if we
are interested in the weakly perturbed case, Eq. (2.4), we only need the functions Z
and I. It can be shown that these quantities are given by 2π-periodic solutions to the
following adjoint linear ODEs [3, 5, 16, 24, 30, 38]:
ω
d
dθ
Z(θ) = −J (θ)
Z(θ), ω
d
dθ
I(θ) = −[J (θ)
− λ]I(θ),
(2.7)