16
H. Nakao
(a)
(b)
(c)
(d)
0.96
-1.4
0.32
-0.62
2
0
0
80
0
80
0
2
x
x
60
80
x
-2
-1
0
1
2
u
0
2 0
4 0
0
2 0
4 0
60
80
x
-0.5
0
0.5
v
Fig. 2.2 Oscillating-spot solution of the FitzHugh-Nagumo model. a, b Spatial profiles of the u
and v components, X(x, 0) = (u(x, 0), v(x, 0)) , at θ = 0. c, d One-period evolution of the u and
v components, X(x, θ) = (u(x, θ), v(x, θ)) , for 0 ≤ θ < 2π
assumption excludes continuous translational symmetries other than the temporal
one along the limit cycle. Thus, the system possesses only a single phase variable.
Typical examples of limit-cycle solutions of the RD systems are traveling pulses
on a ring, oscillating spots, target patterns, and spiral waves in spatially one or twodimensional systems [15, 25]. Figure 2.2 shows the oscillating-spot solution of the
FitzHugh-Nagumo (FHN) model on a 1-dimensional interval, where x ∈ [0, L] with
L = 80 represents the spatial position instead of the vector x (see Sect. 2.5 for details).
Such complex oscillatory patterns in RD systems are difficult to analyze because of
their nonlinearity and infinite-dimensionality. However, if we are interested in the
vicinity of the limit cycle, namely, if we focus on the cases that the oscillatory patterns
are only weakly perturbed, we can approximately describe their infinite-dimensional
dynamics by simple finite-dimensional phase-amplitude equations in a similar way
to the case of ODEs. In what follows, generalizing the method of phase reduction
formulated in Ref. [25], we formulate a method of phase-amplitude reduction for
RD systems exhibiting stable limit-cycle oscillations.
To this end, we need to introduce the phase θ and amplitude r of the infinitedimensional state X(·, t) of the RD system. Because we map the field variable to
scalars, they should be given by functionals of X(·, t). We thus define them as θ(t) =
t)] and r (t) = R[X(·, t)], where : B → [0, 2π) is the phase functional and
R : B → R is the amplitude functional, respectively. Here, we again focus on the
slowest-decaying amplitude associated with the largest non-zero Floquet exponent
λ (< 0), which we assume simple, real, and distant from 0 with a finite spectral gap.
As in the ODE case, we require that these θ and r obey simple equations, i.e.,
˙
θ = ω and ˙
r = λr . Then, from the chain rule of the derivative of functionals,
H. Nakao
(a)
(b)
(c)
(d)
0.96
-1.4
0.32
-0.62
2
0
0
80
0
80
0
2
x
x
60
80
x
-2
-1
0
1
2
u
0
2 0
4 0
0
2 0
4 0
60
80
x
-0.5
0
0.5
v
Fig. 2.2 Oscillating-spot solution of the FitzHugh-Nagumo model. a, b Spatial profiles of the u
and v components, X(x, 0) = (u(x, 0), v(x, 0)) , at θ = 0. c, d One-period evolution of the u and
v components, X(x, θ) = (u(x, θ), v(x, θ)) , for 0 ≤ θ < 2π
assumption excludes continuous translational symmetries other than the temporal
one along the limit cycle. Thus, the system possesses only a single phase variable.
Typical examples of limit-cycle solutions of the RD systems are traveling pulses
on a ring, oscillating spots, target patterns, and spiral waves in spatially one or twodimensional systems [15, 25]. Figure 2.2 shows the oscillating-spot solution of the
FitzHugh-Nagumo (FHN) model on a 1-dimensional interval, where x ∈ [0, L] with
L = 80 represents the spatial position instead of the vector x (see Sect. 2.5 for details).
Such complex oscillatory patterns in RD systems are difficult to analyze because of
their nonlinearity and infinite-dimensionality. However, if we are interested in the
vicinity of the limit cycle, namely, if we focus on the cases that the oscillatory patterns
are only weakly perturbed, we can approximately describe their infinite-dimensional
dynamics by simple finite-dimensional phase-amplitude equations in a similar way
to the case of ODEs. In what follows, generalizing the method of phase reduction
formulated in Ref. [25], we formulate a method of phase-amplitude reduction for
RD systems exhibiting stable limit-cycle oscillations.
To this end, we need to introduce the phase θ and amplitude r of the infinitedimensional state X(·, t) of the RD system. Because we map the field variable to
scalars, they should be given by functionals of X(·, t). We thus define them as θ(t) =
t)] and r (t) = R[X(·, t)], where : B → [0, 2π) is the phase functional and
R : B → R is the amplitude functional, respectively. Here, we again focus on the
slowest-decaying amplitude associated with the largest non-zero Floquet exponent
λ (< 0), which we assume simple, real, and distant from 0 with a finite spectral gap.
As in the ODE case, we require that these θ and r obey simple equations, i.e.,
˙
θ = ω and ˙
r = λr . Then, from the chain rule of the derivative of functionals,