164
Z. Zheng
Eq. (4.45) represents the interaction between two oscillators. By introducing the
polar coordinates r 1,2 (t) and θ 1,2 (t) as
z 1,2 (t) = r 1,2 (t)e
iθ 1,2 (t)
,
(4.46)
the motion in two complex equations of (4.45) can be decomposed into four real
equations of the amplitudes and phases as
˙
r 1,2 (t) = λ 1,2 r 1,2 (t) − b 1,2 r
3
1,2 + K 1,2 r
3
2,1 Re[e
i(θ 2,1 −θ 1,2 )
],
(4.47a)
˙
θ 1,2 (t) = ω 12 + K 1,2 (r
3
2,1 /r 1,2 )Im[e
i(θ 2,1 −θ 1,2 )
].
(4.47b)
It can be seen from (4.47b) that the coupling term will adapt the actual phase
velocities ˙
θ 1,2 (t) even if two oscillators have different natural frequency ω 1,2 .
We consider the possibility of the attracting tendency of two oscillators in the
presence of coupling. By comparing the coupling terms in (4.47a) and (4.47b), it can
be found that the first one is of order r
3 , while the latter is of order r
2 . Therefore
the coupling term in (4.47a) can be neglected, and the two equations in (4.47a) are
decoupled and can be solved. As t → ∞, r 1,2 → r 10,20 =
λ 1,2 /b 1,2 . By substituting
the amplitudes r 1,2 in (4.47b), one can obtain the following coupled phase equations:
˙
θ 1,2 (t) = ω 1,2 + K 1,2 (
r
3
20,10
r 10,20
) sin(θ 2,1 − θ 1,2 ).
(4.48)
The above procedure implies the separation of time scales of the amplitude and
phase, which is actually the consequence of the slaving principle by setting ˙
r 1,2 =
0, i.e. the amplitudes r 1,2 (t) are fast state variables, while the phases θ 1,2 (t) are
neutrally-stable slow variables. Let us keep in working out the phase dynamics of
two coupled oscillators. By introducing the phase difference θ(t) = θ 2 (t) − θ 1 (t)
and natural-frequency difference = ω 2 − ω 1 , Eq. (4.48) can be changed to
˙
θ = − α sin θ,
(4.49)
where the parameter
α = K 2 (
r
3
10
r 20
) + K 1 (
r
3
20
r 10
).
The evolution of (4.49) can be easily solved by integrating the differential
equations and eventually one gets
Z. Zheng
Eq. (4.45) represents the interaction between two oscillators. By introducing the
polar coordinates r 1,2 (t) and θ 1,2 (t) as
z 1,2 (t) = r 1,2 (t)e
iθ 1,2 (t)
,
(4.46)
the motion in two complex equations of (4.45) can be decomposed into four real
equations of the amplitudes and phases as
˙
r 1,2 (t) = λ 1,2 r 1,2 (t) − b 1,2 r
3
1,2 + K 1,2 r
3
2,1 Re[e
i(θ 2,1 −θ 1,2 )
],
(4.47a)
˙
θ 1,2 (t) = ω 12 + K 1,2 (r
3
2,1 /r 1,2 )Im[e
i(θ 2,1 −θ 1,2 )
].
(4.47b)
It can be seen from (4.47b) that the coupling term will adapt the actual phase
velocities ˙
θ 1,2 (t) even if two oscillators have different natural frequency ω 1,2 .
We consider the possibility of the attracting tendency of two oscillators in the
presence of coupling. By comparing the coupling terms in (4.47a) and (4.47b), it can
be found that the first one is of order r
3 , while the latter is of order r
2 . Therefore
the coupling term in (4.47a) can be neglected, and the two equations in (4.47a) are
decoupled and can be solved. As t → ∞, r 1,2 → r 10,20 =
λ 1,2 /b 1,2 . By substituting
the amplitudes r 1,2 in (4.47b), one can obtain the following coupled phase equations:
˙
θ 1,2 (t) = ω 1,2 + K 1,2 (
r
3
20,10
r 10,20
) sin(θ 2,1 − θ 1,2 ).
(4.48)
The above procedure implies the separation of time scales of the amplitude and
phase, which is actually the consequence of the slaving principle by setting ˙
r 1,2 =
0, i.e. the amplitudes r 1,2 (t) are fast state variables, while the phases θ 1,2 (t) are
neutrally-stable slow variables. Let us keep in working out the phase dynamics of
two coupled oscillators. By introducing the phase difference θ(t) = θ 2 (t) − θ 1 (t)
and natural-frequency difference = ω 2 − ω 1 , Eq. (4.48) can be changed to
˙
θ = − α sin θ,
(4.49)
where the parameter
α = K 2 (
r
3
10
r 20
) + K 1 (
r
3
20
r 10
).
The evolution of (4.49) can be easily solved by integrating the differential
equations and eventually one gets