IEEE Circuits and Systems Magazine - Q3 2019 - 13

In 2003, Jadbabie et  al. proposed a criterion about
the synchronization of birds velocities  [72]. To be concrete, if we divide the whole time of birds' collective
motion into an infinite sequence of bounded, non-overlapping, contiguous and nonempty time-intervals, and
make sure that the topology of birds or agents is jointly
connected, then the birds flock will reach consensus
eventually  [72]. In 2007, in order to overcome the difficulty of requirement of being jointly connected, Tang
and Guo introduced a criterion under the framework of
stochastic process  [73]. Suppose that the initial position and the headings of n birds in a flock are all obey
the conditions of independent and identical uniform
distribution on [0, 1]2, the linearized Vicsek model of
system will be synchronized and the corresponding
6
probability will be larger than 1 - O (n -n/log n ) for sufficiently large n and arbitrary radius  [73]. Chen and
other coauthors established a further work of flocks
consensus in 2012 [74]. They revealed that, in a certain
sense, in order to make sure the overall synchronization of flocks, the smallest possible interaction radius
would approximately equal to log n/(rn) with n being
the flock size. This result coincides with the critical radius for connectivity of random geometric graphs given
by Gupta and Kumar [75]. The consensus of bird flocks
is figured as Fig. 7.
b) The Cucker-Smale Model
In the aforementioned Vicsek model, the velocity of bird
i (i ! I ) has a constant magnitude, meanwhile adjusts
its heading to the average of neighbors' headings, while
many researchers hope that there exists a more dynamical model to meet the more sophisticated demand. In
2007, Cucker and Smale established the Cucker-Smale
model  [76]-[77]. Following the thought of Cucker and
Smale, they let one consider n birds moving in a three
dimensional space, and endeavoring to flocking together. Let A(t ) = (aij (t )) represents the changing adjacency
matrix among birds, in which
a ij (t ) =

Under the continuous time condition, let Tt tend to
zero, thus one can obtain the equations of bird flocks
'

xo = v
vo =-Lv.

(6)

Since Eq. (6) has a unique solution, which can be stated as follows [76]-[77]:
1) If b $ 1/2, it is possible that some birds would be
pushed astray or even the whole flock would be
split up. However, if the initial velocities and positions of birds are under certain conditions, birds
flocking would occur in finite time.
2) If b # 1/2, when time t " 3, for arbitrary bird i
and j (i, j ! I ), the relative position will remain
bounded, that is to say, vector x i - x j will tend to
a limit vector x ij and the velocity v i (t ) will tend to
a constant limit v ). Thereby, collision avoidance
and velocity matching are satisfied, and thus the
birds flocking will occur.
Up to now, researchers have modified the CuckerSmale model by adding various forms of perturbations
and obtained similar results as the original model [78]-
[80]. In 2009, Ha et  al. proposed a stochastic CuckerSmale model, and found that bird flocks are independent
of the initial states when the communication rate among
birds is constant [79]. In 2010, Cucker and Dong extended the Cucker-Smale model by adding a repelling force
between birds, and showed that in this case, collision
avoidance and birds flocking will be ensured [81].
Researchers also wanted to excavate more features
from the Cucker-Smale model in different versions. For instance, some researchers have suggested that the flocking of the Cucker-Smale model would exhibit exponential
convergence using the mean field limit method. [82]-[83].
And some others devoted themselves to a class of kinetic
flocks using the Cucker-Smale model [84]-[88], including

1
(1 + < x i (t ) - x j (t )<2 ) b

describes the communication between bird i and j
(i, j ! I ), and b 2 0 is a system parameter which effects
the influence strength. While the Laplacian matrix is given by L = D - A, where D is a diagonal matrix, of which
the ith diagonal element is defined by d i = / nj = 1 a ij .
Cucker and Smale ignored the noise and assumed
that the updating law of velocity is
n

v i (t + Tt ) = v i (t ) + Tt / a ij (v j (t ) - v i (t )).
j =1

THIRD QUARTER 2019

(5)

Figure 7. Consensus of bird flocks [132].

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IEEE Circuits and Systems Magazine - Q3 2019

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