166
Z. Zheng
4.4.2.2 Synchronization Bifurcation Tree
For N > 2 coupled oscillators, it is not an easy task to give analytical discussions of
the microdynamics of synchronization. One usually performs numerical simulations
and compute some useful quantities. Let us consider the following nearest-neighbor
coupled oscillators:
˙
θ i = ω i +
K
3
[sin(θ i+1 − θ i ) + sin(θ i−1 − θ i )],
(4.55)
where i = 1, 2, . . . , N , {ω i } are natural frequencies of oscillators, K is the coupling
strength. Without losing generality, we assume that
i ω i = 0.
When the coupling strength K is increased from 0, different from the two-oscillator
case, the system will show complicated synchronization dynamics because of the
competition between the ordering induced by the coupling and the disorder of natural
frequencies. For the nearest-neighbor coupling case, there is an additional competition, i.e. the competition between the coupling distance and the natural-frequency
differences. As the coupling strength increases, the system will gradually reach the
global synchronization. There is a critical coupling K c , when K > K c the frequencies
of all the oscillators are locked to each other. As K < K c , a portion of oscillators
are synchronized, which is called partial synchronization. To observe the synchrony
process, we define the average frequency of the i-th oscillator as
ω i = lim
T →∞
1
T
T
0
˙
θ i (t)dt.
(4.56)
Synchronization between the i-th and the j-th oscillators is achieved when
−
ω i =
−
ω j .
As the coupling strength changes, oscillators will undergo a coordinated process to
achieve global synchronization.
To observe the synchronization of multiple oscillators clearly, we introduced the
so-called synchronization bifurcation tree (SBT), which is defined as the set of the
relation {
−
ω i (K)}, i.e. the relationship of the average frequencies of all oscillators
and the coupling strength K. The SBT method gives a tree-structured process of
synchronization transitions and exhibits vividly how oscillators are organized to
become synchronized by varying the coupling [96–99].
In Fig. 4.9a–b, we plot the average frequencies {
−
ω i } defined in Eq. (4.56) against
the coupling strength K for N = 5 and 15, respectively, by varying K from K = 0
toK = K c . In both figures, we find interesting transition trees of synchronizations.
When K = 0, all oscillators have different winding numbers, and an increase of the
coupling may lead to a merging of
−
ω i . As two oscillators become synchronous with
each other, their frequencies become the same at a critical coupling and keep the
same (a single curve) with further increase of the coupling strength.
Z. Zheng
4.4.2.2 Synchronization Bifurcation Tree
For N > 2 coupled oscillators, it is not an easy task to give analytical discussions of
the microdynamics of synchronization. One usually performs numerical simulations
and compute some useful quantities. Let us consider the following nearest-neighbor
coupled oscillators:
˙
θ i = ω i +
K
3
[sin(θ i+1 − θ i ) + sin(θ i−1 − θ i )],
(4.55)
where i = 1, 2, . . . , N , {ω i } are natural frequencies of oscillators, K is the coupling
strength. Without losing generality, we assume that
i ω i = 0.
When the coupling strength K is increased from 0, different from the two-oscillator
case, the system will show complicated synchronization dynamics because of the
competition between the ordering induced by the coupling and the disorder of natural
frequencies. For the nearest-neighbor coupling case, there is an additional competition, i.e. the competition between the coupling distance and the natural-frequency
differences. As the coupling strength increases, the system will gradually reach the
global synchronization. There is a critical coupling K c , when K > K c the frequencies
of all the oscillators are locked to each other. As K < K c , a portion of oscillators
are synchronized, which is called partial synchronization. To observe the synchrony
process, we define the average frequency of the i-th oscillator as
ω i = lim
T →∞
1
T
T
0
˙
θ i (t)dt.
(4.56)
Synchronization between the i-th and the j-th oscillators is achieved when
−
ω i =
−
ω j .
As the coupling strength changes, oscillators will undergo a coordinated process to
achieve global synchronization.
To observe the synchronization of multiple oscillators clearly, we introduced the
so-called synchronization bifurcation tree (SBT), which is defined as the set of the
relation {
−
ω i (K)}, i.e. the relationship of the average frequencies of all oscillators
and the coupling strength K. The SBT method gives a tree-structured process of
synchronization transitions and exhibits vividly how oscillators are organized to
become synchronized by varying the coupling [96–99].
In Fig. 4.9a–b, we plot the average frequencies {
−
ω i } defined in Eq. (4.56) against
the coupling strength K for N = 5 and 15, respectively, by varying K from K = 0
toK = K c . In both figures, we find interesting transition trees of synchronizations.
When K = 0, all oscillators have different winding numbers, and an increase of the
coupling may lead to a merging of
−
ω i . As two oscillators become synchronous with
each other, their frequencies become the same at a critical coupling and keep the
same (a single curve) with further increase of the coupling strength.