188
Z. Zheng
(1) Synchronous State:
Fixed points (x 3,4 , y 3,4 ) satisfy |z| = 1, which correspond to the synchronous
state(SS) of the star, i.e. the phase difference between the hub and leaves keeps
constant:
ϕ j (t) = const, j = 1, 2, . . . , N .
(4.143)
It can be seen from Table 4.1 that (x 4 , y 4 ) is always unstable, whilst the state
(x 3 , y 3 ) is stable in the region shown in Table 4.1.
(2) Splay State:
When the modulo of the fixed points
x 1,2 , y 1,2
satisfy |z| > 1, these points
are unphysical. Only when |z| < 1 the fixed points
x 1,2 , y 1,2
are related to the
collective splay state (SPS), i.e.
ϕ j (t) = ϕ(t + jT /N ), j = 1, 2, . . . , N ,
(4.144)
where T is the period of ϕ(t). This state indicates that leaf oscillators can achieve
an ordered state with a fixed time delay between neighboring leaves. All possible
stable regions of the SPS are given in Table 4.1.
4.4.6.3 Periodic Synchronous States
It should be emphasized that the long-term solutions of (4.140) include not only the
stationary states (x 1∼4 , y 1∼4 ), but also time-dependent states. Because the complex
Eq. (4.139) is two-dimensional in real space, the possible time-dependent solution
should be periodic. Further analysis indicates that the system possesses two types
of periodic solutions, where type I exists in the parameter region 0 < α < π/2
and K < K ec = K 2 , and the type-II periodic solution can be found for some special
values of frustrations such as α = 0, ±π/2. These two types of periodic dynamics
are different. We give a brief discussion of these two solutions.
(1) In-phase State:
Type-I periodic solution can be found by using the polar coordinate z = Re
i ,
and Eqs. (4.140) can be transformed to
˙
R = −
K
2
(R
2
− 1) cos(( + α),
˙
= −
K
2
(R +
1
R
) sin(( − α) + ω − NKR sin(( + α).
(4.145)
Equations (4.145) have a limit-cycle solution with a radius R = 1 and a cyclic
evolution of the phase variable (t). This periodic solution is related to the
so-called in-phase state (IPS), where phases of all oscillators are synchronous
as.
Z. Zheng
(1) Synchronous State:
Fixed points (x 3,4 , y 3,4 ) satisfy |z| = 1, which correspond to the synchronous
state(SS) of the star, i.e. the phase difference between the hub and leaves keeps
constant:
ϕ j (t) = const, j = 1, 2, . . . , N .
(4.143)
It can be seen from Table 4.1 that (x 4 , y 4 ) is always unstable, whilst the state
(x 3 , y 3 ) is stable in the region shown in Table 4.1.
(2) Splay State:
When the modulo of the fixed points
x 1,2 , y 1,2
satisfy |z| > 1, these points
are unphysical. Only when |z| < 1 the fixed points
x 1,2 , y 1,2
are related to the
collective splay state (SPS), i.e.
ϕ j (t) = ϕ(t + jT /N ), j = 1, 2, . . . , N ,
(4.144)
where T is the period of ϕ(t). This state indicates that leaf oscillators can achieve
an ordered state with a fixed time delay between neighboring leaves. All possible
stable regions of the SPS are given in Table 4.1.
4.4.6.3 Periodic Synchronous States
It should be emphasized that the long-term solutions of (4.140) include not only the
stationary states (x 1∼4 , y 1∼4 ), but also time-dependent states. Because the complex
Eq. (4.139) is two-dimensional in real space, the possible time-dependent solution
should be periodic. Further analysis indicates that the system possesses two types
of periodic solutions, where type I exists in the parameter region 0 < α < π/2
and K < K ec = K 2 , and the type-II periodic solution can be found for some special
values of frustrations such as α = 0, ±π/2. These two types of periodic dynamics
are different. We give a brief discussion of these two solutions.
(1) In-phase State:
Type-I periodic solution can be found by using the polar coordinate z = Re
i ,
and Eqs. (4.140) can be transformed to
˙
R = −
K
2
(R
2
− 1) cos(( + α),
˙
= −
K
2
(R +
1
R
) sin(( − α) + ω − NKR sin(( + α).
(4.145)
Equations (4.145) have a limit-cycle solution with a radius R = 1 and a cyclic
evolution of the phase variable (t). This periodic solution is related to the
so-called in-phase state (IPS), where phases of all oscillators are synchronous
as.