12
H. Nakao
is a 4-dimensional ordinary differential equations (ODEs) with complex nonlinear
terms representing the dynamics of the membrane potential and channel variables [5].
Due to nonlinearity, analytical solutions of limit cycles are rarely available, requiring approximate theoretical approaches to their synchronization dynamics. The phase
reduction [3–5, 7, 15, 16, 21, 24, 27, 31] is a classical and standard theoretical
method for analyzing weakly perturbed limit-cycle oscillators, which approximately
describes the oscillator state by using only a single phase value and simplifies the multidimensional nonlinear dynamical equations of the oscillator to a one-dimensional
phase equation. It has been successfully used in analyzing, e.g., nonlinear waves and
collective oscillations in populations of weakly coupled limit-cycle oscillators.
Recently, the method of phase reduction has been extended in several ways. In
particular, (i) generalization to infinite-dimensional systems such as partial differential equations (PDEs) [25] and delay-differential equations [13, 26] and (ii) inclusion
of amplitude degrees of freedom representing deviations of the system state from
the unperturbed limit cycle [30, 35, 38] have been formulated. The first extension
is important in analyzing collective oscillations that arise e.g. in spatially extended
populations of dynamical units described by PDEs. The second extension, which
gives reduced amplitude equations in addition to the phase equation, is relatively
recent even for ODEs, but it is necessary for describing transient relaxation dynamics of the perturbed system state to the limit cycle and can be used e.g. for stabilizing
the oscillations by introducing feedback control of the amplitudes.
In this chapter, we formulate the method of phase and amplitude reduction for
stable oscillatory patterns arising in spatially extended reaction-diffusion systems.
2.2 Phase-Amplitude Reduction of Limit-Cycle Oscillators
We first review the method of phase-amplitude reduction [30, 35, 38] for finitedimensional limit-cycle oscillators described by ODEs of the form
˙
X(t) = F(X(t)),
(2.1)
where X(t) ∈ R
n is a n-dimensional oscillator state at time t and F : R
n
→ R
n is a
sufficiently smooth vector field representing the system dynamics. We assume that
Eq. (2.1) has an exponentially stable limit-cycle solution X 0 (t) of natural period T
and frequency ω = 2π/T , satisfying X 0 (t + T ) = X 0 (t). We denote this limit-cycle
attractor as χ and its basin of attraction as B ⊆ R
n .
In the conventional method of phase reduction, the essential step is the introduction
of the asymptotic phase [39] (see Fig. 2.1 for a schematic diagram). Namely, we
assign a scalar phase value θ ∈ [0, 2π) to the oscillator state X ∈ B which eventually
converges to the limit cycle χ. We denote this assignment as θ = (X), where
: B → [0, 2π) is a phase function, and require that the phase θ always increases
with a constant frequency ω as the oscillator state X evolves according to Eq. (2.1),
namely, the function satisfies
H. Nakao
is a 4-dimensional ordinary differential equations (ODEs) with complex nonlinear
terms representing the dynamics of the membrane potential and channel variables [5].
Due to nonlinearity, analytical solutions of limit cycles are rarely available, requiring approximate theoretical approaches to their synchronization dynamics. The phase
reduction [3–5, 7, 15, 16, 21, 24, 27, 31] is a classical and standard theoretical
method for analyzing weakly perturbed limit-cycle oscillators, which approximately
describes the oscillator state by using only a single phase value and simplifies the multidimensional nonlinear dynamical equations of the oscillator to a one-dimensional
phase equation. It has been successfully used in analyzing, e.g., nonlinear waves and
collective oscillations in populations of weakly coupled limit-cycle oscillators.
Recently, the method of phase reduction has been extended in several ways. In
particular, (i) generalization to infinite-dimensional systems such as partial differential equations (PDEs) [25] and delay-differential equations [13, 26] and (ii) inclusion
of amplitude degrees of freedom representing deviations of the system state from
the unperturbed limit cycle [30, 35, 38] have been formulated. The first extension
is important in analyzing collective oscillations that arise e.g. in spatially extended
populations of dynamical units described by PDEs. The second extension, which
gives reduced amplitude equations in addition to the phase equation, is relatively
recent even for ODEs, but it is necessary for describing transient relaxation dynamics of the perturbed system state to the limit cycle and can be used e.g. for stabilizing
the oscillations by introducing feedback control of the amplitudes.
In this chapter, we formulate the method of phase and amplitude reduction for
stable oscillatory patterns arising in spatially extended reaction-diffusion systems.
2.2 Phase-Amplitude Reduction of Limit-Cycle Oscillators
We first review the method of phase-amplitude reduction [30, 35, 38] for finitedimensional limit-cycle oscillators described by ODEs of the form
˙
X(t) = F(X(t)),
(2.1)
where X(t) ∈ R
n is a n-dimensional oscillator state at time t and F : R
n
→ R
n is a
sufficiently smooth vector field representing the system dynamics. We assume that
Eq. (2.1) has an exponentially stable limit-cycle solution X 0 (t) of natural period T
and frequency ω = 2π/T , satisfying X 0 (t + T ) = X 0 (t). We denote this limit-cycle
attractor as χ and its basin of attraction as B ⊆ R
n .
In the conventional method of phase reduction, the essential step is the introduction
of the asymptotic phase [39] (see Fig. 2.1 for a schematic diagram). Namely, we
assign a scalar phase value θ ∈ [0, 2π) to the oscillator state X ∈ B which eventually
converges to the limit cycle χ. We denote this assignment as θ = (X), where
: B → [0, 2π) is a phase function, and require that the phase θ always increases
with a constant frequency ω as the oscillator state X evolves according to Eq. (2.1),
namely, the function satisfies