4 An Introduction to Emergence Dynamics in Complex Systems
147
To obtain an analytical solution, it is convenient to introduce the polar coordinates.
z(t) = r(t)e
iθ(t)
,
and Eq. (4.28) can be decomposed into the following two-dimensional equations of
the phase and the amplitude that are uncoupled to each other:
˙
θ(t) = ω,
(4.29a)
˙
r(t) = λr − br
3
.
(4.29b)
The phase Eq. (4.29a) indicates that the phase evolves uniformly with the phase
velocity ω. The amplitude Eq. (4.29b) has two stationary solutions for λ > 0: the
unstable solution r = 0 and the stable solution r 0 =
√ λ/b. By considering the
dynamics of both the phase and the amplitude, the latter one represents the periodic
solution of Eq. (4.28) as.
z(t → ∞) = r 0 e
i(ωt+θ 0 )
.
This is a stable and attractive limit-cycle, and the relaxation process from an
arbitrary initial state can lead to this solution. The stability of this sustained oscillator
is the result of the competition between the positive feedback term λr and the negative
feedback term −br
3 .
The second source of the feedback mechanism comes from the collaboration of
units in a complex system, which is our focus here. In the following discussions,
we will study the collective oscillation of a population of interacting non-oscillatory
units. Because each unit in the system does not exhibit oscillatory behavior, the
feedback mechanism of sustained oscillations should come from the collaborative
feedback of units.
The self-sustained oscillation in complex systems consisting of a large number
of units is a typical emergence that results from more complicated competitions
and self organizations, and the mechanism of this collective oscillation was of great
interest in recent years [45–47]. It is thus very interesting and important to explore
the mechanism of oscillatory dynamics when these units interact with each other
and study how a number of non-oscillatory nodes organize themselves to emerge a
collective oscillatory phenomenon [48].
4.3.1.2 Self-sustained Oscillations in Complex Systems
The topic of self-sustained oscillations in complex systems was largely motivated by an extensive study and a strong background of system biology. Nonoscillatory systems exist ubiquitously in biological systems, e.g. the gene segments
and neurons. People have extensively studied such common behaviors in nature, such
147
To obtain an analytical solution, it is convenient to introduce the polar coordinates.
z(t) = r(t)e
iθ(t)
,
and Eq. (4.28) can be decomposed into the following two-dimensional equations of
the phase and the amplitude that are uncoupled to each other:
˙
θ(t) = ω,
(4.29a)
˙
r(t) = λr − br
3
.
(4.29b)
The phase Eq. (4.29a) indicates that the phase evolves uniformly with the phase
velocity ω. The amplitude Eq. (4.29b) has two stationary solutions for λ > 0: the
unstable solution r = 0 and the stable solution r 0 =
√ λ/b. By considering the
dynamics of both the phase and the amplitude, the latter one represents the periodic
solution of Eq. (4.28) as.
z(t → ∞) = r 0 e
i(ωt+θ 0 )
.
This is a stable and attractive limit-cycle, and the relaxation process from an
arbitrary initial state can lead to this solution. The stability of this sustained oscillator
is the result of the competition between the positive feedback term λr and the negative
feedback term −br
3 .
The second source of the feedback mechanism comes from the collaboration of
units in a complex system, which is our focus here. In the following discussions,
we will study the collective oscillation of a population of interacting non-oscillatory
units. Because each unit in the system does not exhibit oscillatory behavior, the
feedback mechanism of sustained oscillations should come from the collaborative
feedback of units.
The self-sustained oscillation in complex systems consisting of a large number
of units is a typical emergence that results from more complicated competitions
and self organizations, and the mechanism of this collective oscillation was of great
interest in recent years [45–47]. It is thus very interesting and important to explore
the mechanism of oscillatory dynamics when these units interact with each other
and study how a number of non-oscillatory nodes organize themselves to emerge a
collective oscillatory phenomenon [48].
4.3.1.2 Self-sustained Oscillations in Complex Systems
The topic of self-sustained oscillations in complex systems was largely motivated by an extensive study and a strong background of system biology. Nonoscillatory systems exist ubiquitously in biological systems, e.g. the gene segments
and neurons. People have extensively studied such common behaviors in nature, such