IEEE Circuits and Systems Magazine - Q3 2019 - 18

Noether's Theorem notes that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The spin conservation law
holds if and only if the derivatives of { over t appears
in the motion equation. However, the motion equation
(11) contains a dissipation term {o , thus the conservation breaks. The second confusion is the strange result
of Eq. (11) so that the social force Fs = a 2 J d 2 { = {o , which
implies that Fs controls {o other than {p , this phenomenon disobeys the common sense.
To surmount these obstacles, they added a kinetic
term s 2z /2| to the Hamiltonian (10), where s z is the momentum canonical conjugated to {, which is the intrinsic spin of bird and is directly proportional to curvature,
while | is the generalized moment of inertia, which is
defined as the ratio between the social force (the cause)
and the change of angular velocity (the effect). Then one
can rewrite the Hamiltonian (10) as
H=

#

dx 3 ) 1 a 2 J [d{ (x, t )]2 + s z (x, t ) 3.
2|
a3 2
2

(12)

The equations of motion generated by Hamiltonian (12) are
{o =

2H = s z , s. =- 2H = a 2 J d 2 {.
z
2s z
2{
|

(13)

By taking derivative with respect to time on both sides
of Eq. (13), one obtains
a 2 J d 2 { = 0,
{p |

aJ d 2 { = 0,
|

IEEE CIRCUITS AND SYSTEMS MAGAZINE

xo i = f (x i) +

N

/

c ij (t ) A (t ) (x j - x t ), i ! I,

(15)

(14)

that is, ~ = c s k, c 2s = a 2 J/|. Hence, the information propagates linearly, which is well matched with the empirical
data observed in bird flocks [98].
If the rotation of velocity vectors is described by the
orbital angle, birds' trajectories in a turn are parallel
and have different curvatures. In contrast, if the rotation of velocity vectors is described by phase {, then
all birds' trajectories in a turn share the same curvature. From a biological viewpoint, the former parallel
path turning mode indicates that the birds flying on the
outer side need a sufficiently larger speed than the inside flock members, while the latter maintains the same
speed over the whole flock, it is exactly the way that real
flocks act in turning [98].
Interestingly, the authors showed that the new model
is mathematically equivalent to the equations which describe the lattice-gas of superfluid helium  [98]. These
results contrast starkly with the existing models that
18

C. Collective Decision Making in Control Theory
Over the last decades, consensus of multi-agent systems
(MASs) as one of the most typical collective decision making behaviors, which has attracted extensive attention in
various disciplines, such as control theory  [99],  [120],
physics  [48] and artificial intelligence  [59]. Specifically,
there are huge amount of studies and applications about
consensus based on control theory.
It is worth mentioning that there are some general methods to analyze the consensus behaviors of flocks [100]-
[105], a recently published paper by Chen et al. provided
an excellent review on this topic [112].
Some approaches dealing with continuous-time MASs,
such as Lyapunov functions [106], [120], have already
been well-developed. To investigate the consensus of
continuous time complex networks, in 2005, Lü and Chen
considered a dynamical network consisting of N linearly
and diffusively coupled identical nodes or agents [120].
The proposed general time-varying dynamical network
which is described by

j = 1, j ! i

the corresponding dispersion relation is
~ 2 - 0 · i~ -

describe collective movement, and elucidate that the
Vicsek model is no longer applicable for the description
of birds' turning. "This new theory could apply to other
types of group behavior, such as fish schools or assemblages of moving cells," Professor David Sumpter said.

where x i is the state variable of agent i (i ! I ), A(t ) =
(aij (t )) represents the inner-coupling matrix of the
network at time t, and C (t ) = (c ij (t )) is the coupling configuration matrix which represents the outer-coupling
strength and the topological structure of the network
at time t, in which c ij (t ) is defined just as in the Vicsek
model aforementioned. If A(t ) and C (t ) are constant
matrices, then this network is a time-invariant dynamical network.
The solution of Eq. (15) is denoted by x i (t, x 0), while
s(t, x 0) is a solution of the system xo = f (x) with x 0 as
the initial value. Complex network is said to achieve
consensus if
lim
< x i (t, x 0) - s (t, x 0)< 2 = 0, i ! I.
t"3

(16)

They indicated that the factors which determine
the consensus of a time-varying dynamical network
are inner-coupling matrix, the eigenvalues and the corresponding eigenvectors of the coupling configuration
matrix [120]. This novel result is different from those
for consensus of time-invariant complex networks acquired previously.
THIRD QUARTER 2019
IEEE Circuits and Systems Magazine - Q3 2019

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q3 2019
