12 Phase Reconstruction with Iterated Hilbert Transforms
193
12.2 Nonlinear Oscillators and Phase Reduction
Here we briefly review the phase reduction of driven limit cycle oscillators, for more
details see [26, 27]. An autonomous oscillator is described by N state variables y
which evolve according to a system of differential equations ˙
y = f(y). One assumes
that this system has a stable limit cycle y 0 (t) = y 0 (t + T ) describing periodic (period
T ) oscillations. In the basin of attraction of the cycle one can always introduce a phase
variable ϕ which grows uniformly in time
˙
ϕ = ω =
2π
T
.
(12.1)
On the limit cycle, only the phase varies, so that y 0 (ϕ) = y 0 (ϕ + 2π), which
means that the value of the phase uniquely determines the point on the limit cycle.
If the autonomous oscillator is perturbed, i.e., it is driven by a small external force
˙
y = f(y) + εp(y, t), then the system slightly (of order ∼ ε) deviates from the limit
cycle, and additionally the phase does not grow uniformly, but obeys (in the first
order in ε) the equation
˙
ϕ = ω + εQ(ϕ, t),
(12.2)
where Q can be expressed via f, p (see [26] for details). Equation (12.2) contains only
the phase and not the amplitude, it can be viewed as a result of the phase reduction.
The dynamics of the phase according to (12.2) allows for studying different important
effects of synchronization, etc. In the case when the oscillator is forced by another
one, the force p(η) can be viewed as a function of the phase η(t) of this driving
oscillator, so the function Q(ϕ, η) becomes the coupling function depending on two
phases. In experimental situations it is quite common to perturb just one variable
of the system. In that case, if the forcing term is scalar and does not depend on the
system variables, one can factorize Q(ϕ, t) = Z (ϕ)P(t) into the iPRC Z (ϕ) and the
(scalar) external driving P(t) [5, 39].
Example: Forced Stuart-Landau Oscillator. In this contribution we consider as
an example the perturbed Stuart-Landau oscillator (SL)
˙
a = (μ + iν)a − (1 + iα)a|a|
2
+ iεP(t),
P(t) = cos(r ωt)
(12.3)
where a(t) := R(t) exp[iφ(t)] is the complex amplitude. Parameter μ determines the
amplitude (
√
μ) and stability of the limit cycles, α is the nonisochronicity parameter.
It is easy to check that
ϕ(t) = φ(t) − α ln[R(t)]
(12.4)
is the proper phase, rotating, independently of amplitude R, with uniform frequency
ω = ν − μα. The frequency of the forcing is r ω, where parameter r is the ratio of
the external frequency to the base frequency ω. In the first order in ε, the amplitude
and the phase dynamics read
193
12.2 Nonlinear Oscillators and Phase Reduction
Here we briefly review the phase reduction of driven limit cycle oscillators, for more
details see [26, 27]. An autonomous oscillator is described by N state variables y
which evolve according to a system of differential equations ˙
y = f(y). One assumes
that this system has a stable limit cycle y 0 (t) = y 0 (t + T ) describing periodic (period
T ) oscillations. In the basin of attraction of the cycle one can always introduce a phase
variable ϕ which grows uniformly in time
˙
ϕ = ω =
2π
T
.
(12.1)
On the limit cycle, only the phase varies, so that y 0 (ϕ) = y 0 (ϕ + 2π), which
means that the value of the phase uniquely determines the point on the limit cycle.
If the autonomous oscillator is perturbed, i.e., it is driven by a small external force
˙
y = f(y) + εp(y, t), then the system slightly (of order ∼ ε) deviates from the limit
cycle, and additionally the phase does not grow uniformly, but obeys (in the first
order in ε) the equation
˙
ϕ = ω + εQ(ϕ, t),
(12.2)
where Q can be expressed via f, p (see [26] for details). Equation (12.2) contains only
the phase and not the amplitude, it can be viewed as a result of the phase reduction.
The dynamics of the phase according to (12.2) allows for studying different important
effects of synchronization, etc. In the case when the oscillator is forced by another
one, the force p(η) can be viewed as a function of the phase η(t) of this driving
oscillator, so the function Q(ϕ, η) becomes the coupling function depending on two
phases. In experimental situations it is quite common to perturb just one variable
of the system. In that case, if the forcing term is scalar and does not depend on the
system variables, one can factorize Q(ϕ, t) = Z (ϕ)P(t) into the iPRC Z (ϕ) and the
(scalar) external driving P(t) [5, 39].
Example: Forced Stuart-Landau Oscillator. In this contribution we consider as
an example the perturbed Stuart-Landau oscillator (SL)
˙
a = (μ + iν)a − (1 + iα)a|a|
2
+ iεP(t),
P(t) = cos(r ωt)
(12.3)
where a(t) := R(t) exp[iφ(t)] is the complex amplitude. Parameter μ determines the
amplitude (
√
μ) and stability of the limit cycles, α is the nonisochronicity parameter.
It is easy to check that
ϕ(t) = φ(t) − α ln[R(t)]
(12.4)
is the proper phase, rotating, independently of amplitude R, with uniform frequency
ω = ν − μα. The frequency of the forcing is r ω, where parameter r is the ratio of
the external frequency to the base frequency ω. In the first order in ε, the amplitude
and the phase dynamics read