2 Phase and Amplitude Description of Complex Oscillatory Patterns …
13
Fig. 2.1 Phase and
amplitude of a limit cycle.
Limit cycle χ (thick black
circle), isochrons (red dashed
curves), and isostables (thin
blue circles). The green dot
shows the oscillator state
X 0 (θ), red arrow the phase
sensitivity function Z(θ), and
the blue arrow the amplitude
sensitivity function I(θ) at
phase θ, respectively
X0(θ)
Z(θ)
I(θ)
ω
˙
θ = ˙
= ∇ X · ˙
X = F(X) · ∇ X = ω
(2.2)
for X ∈ B, where ∇ X ∈ R
n is the gradient vector of and · represents the
ordinary dot product between two vectors. The level sets of are called isochrons.
The phase function satisfying Eq. (2.2) can be obtained as follows. For the
state X 0 (t) at time t started from a reference state X R at time 0 on χ, the phase
function can be taken as 0 (t)) = ωt (mod 2π), where X R gives the origin of the
phase, R ) = 0. This assigns a phase value θ ∈ [0, 2π) to each point on χ. In what
follows, we denote the state on χ as X 0 (θ) as a function of θ. Note that θ = 0 (θ))
holds. To the state X not on χ, we assign a phase value = θ if it converges
to the same state on χ as X 0 (θ), namely, if lim τ →∞
τ X − S
τ X 0 (θ) → 0, where
· · is the Euclidean norm and S
τ represents the time-τ flow of Eq. (2.1), satisfying
S
τ X(t) = X(t + τ ). The phase function defined as above satisfies Eq. (2.2) for
X ∈ B.
Next, we consider the amplitude degrees of freedom, representing deviations of
the state X from the limit cycle χ. The linear stability of χ is characterized by the
Floquet exponents, 0, λ 2 , . . . , λ n in decreasing order of their real parts, where 0 is
associated with the phase direction, namely, the neutral tangential direction along χ,
and the real parts of all other exponents λ 2 , . . . , λ n are negative. For n-dimensional
oscillators, there are generally n − 1 amplitudes associated with λ 2 , . . . , λ n , but we
focus only on the dominant, slowest-decaying amplitude associated with λ 2 and
denote this exponent as λ (< 0), which we assume simple and real for simplicity.
In a similar way to the phase θ, it is useful to assign a scalar amplitude r = R(X)
to the state X ∈ B and assume that r obeys a simple equation, where R : B → R is
an amplitude function. A natural assumption is that r exponentially decays to 0 as
˙
r = λr when X converges to χ. Here, the decay rate is given by the Floquet exponent
λ and r = 0 when X is on χ. Thus, we require R to satisfy R(X 0 (θ)) = 0 and
˙
r = ˙
R(X) = ∇ X R(X) · ˙
X = F(X) · ∇ X R(X) = λR(X) = λr.
(2.3)
13
Fig. 2.1 Phase and
amplitude of a limit cycle.
Limit cycle χ (thick black
circle), isochrons (red dashed
curves), and isostables (thin
blue circles). The green dot
shows the oscillator state
X 0 (θ), red arrow the phase
sensitivity function Z(θ), and
the blue arrow the amplitude
sensitivity function I(θ) at
phase θ, respectively
X0(θ)
Z(θ)
I(θ)
ω
˙
θ = ˙
= ∇ X · ˙
X = F(X) · ∇ X = ω
(2.2)
for X ∈ B, where ∇ X ∈ R
n is the gradient vector of and · represents the
ordinary dot product between two vectors. The level sets of are called isochrons.
The phase function satisfying Eq. (2.2) can be obtained as follows. For the
state X 0 (t) at time t started from a reference state X R at time 0 on χ, the phase
function can be taken as 0 (t)) = ωt (mod 2π), where X R gives the origin of the
phase, R ) = 0. This assigns a phase value θ ∈ [0, 2π) to each point on χ. In what
follows, we denote the state on χ as X 0 (θ) as a function of θ. Note that θ = 0 (θ))
holds. To the state X not on χ, we assign a phase value = θ if it converges
to the same state on χ as X 0 (θ), namely, if lim τ →∞
τ X − S
τ X 0 (θ) → 0, where
· · is the Euclidean norm and S
τ represents the time-τ flow of Eq. (2.1), satisfying
S
τ X(t) = X(t + τ ). The phase function defined as above satisfies Eq. (2.2) for
X ∈ B.
Next, we consider the amplitude degrees of freedom, representing deviations of
the state X from the limit cycle χ. The linear stability of χ is characterized by the
Floquet exponents, 0, λ 2 , . . . , λ n in decreasing order of their real parts, where 0 is
associated with the phase direction, namely, the neutral tangential direction along χ,
and the real parts of all other exponents λ 2 , . . . , λ n are negative. For n-dimensional
oscillators, there are generally n − 1 amplitudes associated with λ 2 , . . . , λ n , but we
focus only on the dominant, slowest-decaying amplitude associated with λ 2 and
denote this exponent as λ (< 0), which we assume simple and real for simplicity.
In a similar way to the phase θ, it is useful to assign a scalar amplitude r = R(X)
to the state X ∈ B and assume that r obeys a simple equation, where R : B → R is
an amplitude function. A natural assumption is that r exponentially decays to 0 as
˙
r = λr when X converges to χ. Here, the decay rate is given by the Floquet exponent
λ and r = 0 when X is on χ. Thus, we require R to satisfy R(X 0 (θ)) = 0 and
˙
r = ˙
R(X) = ∇ X R(X) · ˙
X = F(X) · ∇ X R(X) = λR(X) = λr.
(2.3)