196
E. Gengel and A. Pikovsky
Note that Eq. (12.10) is well defined as by construction, ˙
θ = f (θ) > 0. In the
case one observes driven oscillations, one approximately estimates f (θ) = = ˙
θ, see
[18] for details.
According to the discussion above, one can perform the phase reconstruction of
an observed signal X (t) in two steps:
(i) Find a decomposition X (t) = ˜
S(θ(t)) into a waveform and a protophase, satisfying conditions
(I): ∀t, ˙
θ(t) > 0,
(II): ˜
S(θ) = ˜
S(θ + 2π).
(12.11)
(ii) Perform a transformation from a protophase to the phase, so that the latter grows
on average uniformly in time
(III): ˙
ϕ = const.
(12.12)
Conditions [I, II] ensure that the reconstructed protophase is monotonous and
2π-periodic. Condition [III] selects the phase as a variable uniformly growing in
time, in contrast to other protophases which according to (12.9) grow with a rate
that is protophase-dependent (with 2π-periodicity). Below we discuss in details the
methods allowing for accomplishing steps (i) and (ii).
12.3.2 Embeddings, Hilbert Transform, and
Phase-Amplitude Mixing
The first task, a decomposition into a waveform and a protophase, is trivial, if two
scalar observables {X (t) = X [y 0 (t)], Y (t) = Y [y 0 (t)]} of the oscillator’s dynamics
are available (of course, these observables should be not fully dependent). In this
case, on the {X, Y } plane one observes a closed continuous curve, parametrized by
the phase, and the trajectory rotates along this curve. Any parametrization of the
curve, normalized by 2π, will then provide a protophase as a function of time. After
this, one has only to accomplish the step (ii), i.e. to transform the protophase to the
phase.
An intrinsically non-trivial problem appears, if only one scalar observable, X (t),
is available. The goal is to perform a two-dimensional embedding of the signal X (t),
by generating from it the second variable Y (t). There exist several approaches for
this task. The most popular ones are the delay-embedding Y (t) = X (t − τ ) [16], the
derivative embedding Y (t) = ˙
X (t) [32], and the Hilbert transform (HT) embedding
Y (t) = ˆ
H [X ](t), where (on a finite interval [t 0 , t m ])
ˆ
H [X ](t) :=
p.v.
π
t m
t 0
X (τ )
t − τ
dτ .
(12.13)
E. Gengel and A. Pikovsky
Note that Eq. (12.10) is well defined as by construction, ˙
θ = f (θ) > 0. In the
case one observes driven oscillations, one approximately estimates f (θ) = = ˙
θ, see
[18] for details.
According to the discussion above, one can perform the phase reconstruction of
an observed signal X (t) in two steps:
(i) Find a decomposition X (t) = ˜
S(θ(t)) into a waveform and a protophase, satisfying conditions
(I): ∀t, ˙
θ(t) > 0,
(II): ˜
S(θ) = ˜
S(θ + 2π).
(12.11)
(ii) Perform a transformation from a protophase to the phase, so that the latter grows
on average uniformly in time
(III): ˙
ϕ = const.
(12.12)
Conditions [I, II] ensure that the reconstructed protophase is monotonous and
2π-periodic. Condition [III] selects the phase as a variable uniformly growing in
time, in contrast to other protophases which according to (12.9) grow with a rate
that is protophase-dependent (with 2π-periodicity). Below we discuss in details the
methods allowing for accomplishing steps (i) and (ii).
12.3.2 Embeddings, Hilbert Transform, and
Phase-Amplitude Mixing
The first task, a decomposition into a waveform and a protophase, is trivial, if two
scalar observables {X (t) = X [y 0 (t)], Y (t) = Y [y 0 (t)]} of the oscillator’s dynamics
are available (of course, these observables should be not fully dependent). In this
case, on the {X, Y } plane one observes a closed continuous curve, parametrized by
the phase, and the trajectory rotates along this curve. Any parametrization of the
curve, normalized by 2π, will then provide a protophase as a function of time. After
this, one has only to accomplish the step (ii), i.e. to transform the protophase to the
phase.
An intrinsically non-trivial problem appears, if only one scalar observable, X (t),
is available. The goal is to perform a two-dimensional embedding of the signal X (t),
by generating from it the second variable Y (t). There exist several approaches for
this task. The most popular ones are the delay-embedding Y (t) = X (t − τ ) [16], the
derivative embedding Y (t) = ˙
X (t) [32], and the Hilbert transform (HT) embedding
Y (t) = ˆ
H [X ](t), where (on a finite interval [t 0 , t m ])
ˆ
H [X ](t) :=
p.v.
π
t m
t 0
X (τ )
t − τ
dτ .
(12.13)