2 Phase and Amplitude Description of Complex Oscillatory Patterns …
17
˙
θ(t) = ˙
[X(·, t)] =
V
δ[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx = ω,
(2.9)
˙
r (t) = ˙
R[X(·, t)] =
V
δ R[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx = λr (t)
(2.10)
should hold. Here, ∂X(x, t)/∂t is given by Eq. (2.8) and δ A[X(·)]/δX(x)| X(x)=X(x,t) ∈
R
n (A = , R) represents the functional derivative of the functional A[X] with
respect to X(x) evaluated at X(x) = X(x, t). Such phase and amplitude functionals
can, in principle, be defined as in the ODE case. We denote the system state on χ
as X 0 (·, θ) as a function of the phase θ ∈ [0, 2π). Note that [X 0 (·, θ)] = θ and
R[X 0 (·, θ)] = 0 hold.
Thus, Eqs. (2.2) and (2.3) for ODEs can formally be generalized to spatially
extended RD systems, where the vector gradient is replaced by the functional derivative. We can also interpret R and = e
i as eigenfunctionals of the infinitesimal
Koopman operator for RD systems [22, 23]. We note here that the explicit forms of the
functionals and R, which are difficult to obtain, are not necessary in the following
derivation of the phase and amplitude equations near the limit-cycle solution.
Let us now consider the case that the RD system Eq. (2.8) is weakly perturbed as
∂X(x, t)
∂t
= F(X(x, t); x) + D∇
2 X(x, t) + p(X(·, t), x, t),
(2.11)
where > 0 is a small parameter and p(X(·, t), x, t) ∈ R
n is the applied perturbation
that can depend on the state X, position x, and time t. In a similar way to the case of
ODEs, using the phase and amplitude functionals and R satisfying Eqs. (2.9) and
(2.10), we can reduce Eq. (2.11) to a set of approximate phase-amplitude equations
as
˙
θ =
V
δ[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx ω +
V
Z(x, θ) · p(X 0 (·, θ), x, t)dx,
(2.12)
˙
r =
V
δ R[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx λr +
V
I(x, θ) · p(X 0 (·, θ), x, t)dx,
(2.13)
where ∂X(x, t)/∂t is given by Eq. (2.11). Here, in the last expression in each equation, we have approximately evaluated the functional derivatives and perturbations
in the integral at X 0 (x, θ) rather than at X(x, t), assuming that the system state is
sufficiently close to χ, and introduced the phase and amplitude sensitivity functions
Z(x, θ) =
δ[X(·)]
δX(x)
X(x)=X 0 (x,θ)
, I(x, θ) =
δ R[X(·)]
δX(x)
X(x)=X 0 (x,θ)
,
(2.14)
which now depend also on the position x. The approximate equations (2.12) and
(2.13) are correct up to O() like Eqs. (2.5) and (2.6) in the ODE case.
17
˙
θ(t) = ˙
[X(·, t)] =
V
δ[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx = ω,
(2.9)
˙
r (t) = ˙
R[X(·, t)] =
V
δ R[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx = λr (t)
(2.10)
should hold. Here, ∂X(x, t)/∂t is given by Eq. (2.8) and δ A[X(·)]/δX(x)| X(x)=X(x,t) ∈
R
n (A = , R) represents the functional derivative of the functional A[X] with
respect to X(x) evaluated at X(x) = X(x, t). Such phase and amplitude functionals
can, in principle, be defined as in the ODE case. We denote the system state on χ
as X 0 (·, θ) as a function of the phase θ ∈ [0, 2π). Note that [X 0 (·, θ)] = θ and
R[X 0 (·, θ)] = 0 hold.
Thus, Eqs. (2.2) and (2.3) for ODEs can formally be generalized to spatially
extended RD systems, where the vector gradient is replaced by the functional derivative. We can also interpret R and = e
i as eigenfunctionals of the infinitesimal
Koopman operator for RD systems [22, 23]. We note here that the explicit forms of the
functionals and R, which are difficult to obtain, are not necessary in the following
derivation of the phase and amplitude equations near the limit-cycle solution.
Let us now consider the case that the RD system Eq. (2.8) is weakly perturbed as
∂X(x, t)
∂t
= F(X(x, t); x) + D∇
2 X(x, t) + p(X(·, t), x, t),
(2.11)
where > 0 is a small parameter and p(X(·, t), x, t) ∈ R
n is the applied perturbation
that can depend on the state X, position x, and time t. In a similar way to the case of
ODEs, using the phase and amplitude functionals and R satisfying Eqs. (2.9) and
(2.10), we can reduce Eq. (2.11) to a set of approximate phase-amplitude equations
as
˙
θ =
V
δ[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx ω +
V
Z(x, θ) · p(X 0 (·, θ), x, t)dx,
(2.12)
˙
r =
V
δ R[X(·)]
δX(x)
X(x)=X(x,t)
·
∂X(x, t)
∂t
dx λr +
V
I(x, θ) · p(X 0 (·, θ), x, t)dx,
(2.13)
where ∂X(x, t)/∂t is given by Eq. (2.11). Here, in the last expression in each equation, we have approximately evaluated the functional derivatives and perturbations
in the integral at X 0 (x, θ) rather than at X(x, t), assuming that the system state is
sufficiently close to χ, and introduced the phase and amplitude sensitivity functions
Z(x, θ) =
δ[X(·)]
δX(x)
X(x)=X 0 (x,θ)
, I(x, θ) =
δ R[X(·)]
δX(x)
X(x)=X 0 (x,θ)
,
(2.14)
which now depend also on the position x. The approximate equations (2.12) and
(2.13) are correct up to O() like Eqs. (2.5) and (2.6) in the ODE case.