2 Phase and Amplitude Description of Complex Oscillatory Patterns …
15
where J (θ) = DF(X)| X=X 0 (θ) ∈ R
n×n is a Jacobian matrix of F at X = X 0 (θ) on
χ and denotes transpose. The matrix components of J (θ) are given by J i j (θ) =
∂ F i /∂ X j | X=X 0 (θ) for i, j = 1, . . . , n, where F i and X i are vector components of F
and X, respectively. To be consistent with the definition of the asymptotic phase, Z
should be normalized as Z(θ) · F(X 0 (θ)) = ω for ∀θ ∈ [0, 2π). The normalization of
I, which determines the scale of r , can be chosen arbitrarily. These adjoint equations
can be derived by expanding (X) and R(X) around X 0 (θ) to the first order in
X − X 0 (θ) and plugging them into Eqs. (2.2) and (2.3) [3, 16, 30].
Equations (2.5) and (2.6) give the lowest-order phase-amplitude description of
the weakly perturbed limit-cycle oscillator Eq. (2.4), from which we can predict the
dynamics of the phase θ and amplitude r and in turn use them to predict the oscillator
state near χ.
1 Note that θ is decoupled from r at the lowest-order approximation.
Higher-order approximations can also be developed, which yield coupling between
θ and r and more precisely describe the oscillator dynamics [14, 37]. In this chapter,
we consider the simplest nontrivial lowest-order case and generalize it to PDEs.
2.3 Phase-Amplitude Reduction of Reaction-Diffusion
Systems
We now consider spatially extended reaction-diffusion (RD) systems described by
∂X(x, t)
∂t
= F(X(x, t); x) + D∇
2 X(x, t),
(2.8)
where X(x, t) ∈ R
n is a n-dimensional field variable at position x ∈ V ⊆ R
d in a
d-dimensional spatial domain V and at time t, representing e.g. concentrations of
chemical species involved in the reaction, F : R
n
× R
d
→ R
n represents positiondependent reaction dynamics of X(x, t), D ∈ R
n×n is a matrix of diffusion coefficients, and ∇
2 is a Laplacian operator representing the diffusion of X. Appropriate
boundary conditions, e.g., periodic boundary conditions, are assumed for V . Note
that Eq. (2.8) is an infinite-dimensional dynamical system whose system state at t is
given by X(x, t) for x ∈ V , which we denote as X(·, t) ∈ C, where C is some appropriate space of smooth vector-valued functions. We assume that Eq. (2.8) possesses an
exponentially stable limit-cycle solution X 0 (·, t) of natural period T and frequency
ω = 2π/T , satisfying X 0 (x, t + T ) = X 0 (x, t) for ∀x ∈ V . We again denote this
limit-cycle attractor in C as χ and its basin of attraction as B ⊆ C. Note that this
1 The system state can be approximately represented as X X 0 (θ) + r u(θ) up to O(), where
u(θ) is a 2π-periodic Floquet eigenfunction with Floquet exponent λ satisfying ω(d/dθ)u(θ) =
[J (θ) − λ]u(θ). See e.g. [16, 24] for details. This also holds true for the RD system discussed later
in Sect. 2.3, namely, X(x, t) X 0 (x, θ) + r u(x, θ) where the Floquet eigenfunction u(x, θ) is
2π-periodic in θ and satisfies ω(∂/∂θ)u(x, θ) = [J (x; θ) − λ + D∇ 2 ]u(x, θ).
15
where J (θ) = DF(X)| X=X 0 (θ) ∈ R
n×n is a Jacobian matrix of F at X = X 0 (θ) on
χ and denotes transpose. The matrix components of J (θ) are given by J i j (θ) =
∂ F i /∂ X j | X=X 0 (θ) for i, j = 1, . . . , n, where F i and X i are vector components of F
and X, respectively. To be consistent with the definition of the asymptotic phase, Z
should be normalized as Z(θ) · F(X 0 (θ)) = ω for ∀θ ∈ [0, 2π). The normalization of
I, which determines the scale of r , can be chosen arbitrarily. These adjoint equations
can be derived by expanding (X) and R(X) around X 0 (θ) to the first order in
X − X 0 (θ) and plugging them into Eqs. (2.2) and (2.3) [3, 16, 30].
Equations (2.5) and (2.6) give the lowest-order phase-amplitude description of
the weakly perturbed limit-cycle oscillator Eq. (2.4), from which we can predict the
dynamics of the phase θ and amplitude r and in turn use them to predict the oscillator
state near χ.
1 Note that θ is decoupled from r at the lowest-order approximation.
Higher-order approximations can also be developed, which yield coupling between
θ and r and more precisely describe the oscillator dynamics [14, 37]. In this chapter,
we consider the simplest nontrivial lowest-order case and generalize it to PDEs.
2.3 Phase-Amplitude Reduction of Reaction-Diffusion
Systems
We now consider spatially extended reaction-diffusion (RD) systems described by
∂X(x, t)
∂t
= F(X(x, t); x) + D∇
2 X(x, t),
(2.8)
where X(x, t) ∈ R
n is a n-dimensional field variable at position x ∈ V ⊆ R
d in a
d-dimensional spatial domain V and at time t, representing e.g. concentrations of
chemical species involved in the reaction, F : R
n
× R
d
→ R
n represents positiondependent reaction dynamics of X(x, t), D ∈ R
n×n is a matrix of diffusion coefficients, and ∇
2 is a Laplacian operator representing the diffusion of X. Appropriate
boundary conditions, e.g., periodic boundary conditions, are assumed for V . Note
that Eq. (2.8) is an infinite-dimensional dynamical system whose system state at t is
given by X(x, t) for x ∈ V , which we denote as X(·, t) ∈ C, where C is some appropriate space of smooth vector-valued functions. We assume that Eq. (2.8) possesses an
exponentially stable limit-cycle solution X 0 (·, t) of natural period T and frequency
ω = 2π/T , satisfying X 0 (x, t + T ) = X 0 (x, t) for ∀x ∈ V . We again denote this
limit-cycle attractor in C as χ and its basin of attraction as B ⊆ C. Note that this
1 The system state can be approximately represented as X X 0 (θ) + r u(θ) up to O(), where
u(θ) is a 2π-periodic Floquet eigenfunction with Floquet exponent λ satisfying ω(d/dθ)u(θ) =
[J (θ) − λ]u(θ). See e.g. [16, 24] for details. This also holds true for the RD system discussed later
in Sect. 2.3, namely, X(x, t) X 0 (x, θ) + r u(x, θ) where the Floquet eigenfunction u(x, θ) is
2π-periodic in θ and satisfies ω(∂/∂θ)u(x, θ) = [J (x; θ) − λ + D∇ 2 ]u(x, θ).