# Comment on “An Investigation into the Performance of Particle Swarm Optimization with Various Chaotic Maps”

Authors: Yudong Zhang; Genlin Ji; Zhengchao Dong; Shuihua Wang; Preetha Phillips
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper researched three definitions of Gauss map and found that the definition of “Gauss map” in the paper of Arasomwan and Adewumi may be incoherent with other publications. In addition, we analyzed the difference of continuous Gauss map and the floating-point Gauss map, and we pointed out that the floating-point simulation behaved significantly differently from the continuous Gauss map.


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## Body

## 1. Introduction

Paper [1] is very welcome. The authors investigated the effect of nine chaotic maps on the performance of two variants of particle swarm optimization (PSO) algorithm as random inertia weight PSO (RIW-PSO) and linear decreasing inertia weight PSO (LDIW-PSO). Their simulation results showed that “the performances of those two variants were improved by many of the chaotic maps.”However, the authors give an inappropriate definition of Gauss map (Gaussian map) that some readers may get the mistaken understanding. There are currently three different types of Gauss maps for different disciplines. We will discuss them in what follows.Definition 1.
The first definition is used in differential geometry, where the “Gauss map” maps a surface in Euclidean spaceR
3 to the unit spheres S
2; that is, given a surface X lying in R
3, the Gauss map is a continuous map N
:
X
→
S
2 such that N
(
p
) is a unit vector orthogonal to X at p, namely, the normal vector to X at p.Definition 2.
The second definition of Gauss map is related to continued fractions and is used in programming, chaos, ergodic theory, and so forth. Its form is(1)
f
(
z
)
=
0
,
z
=
0
,
1
z
-
1
z
,
otherwise
,where the  represents the floor function. Note that Corless [2] investigated this Gauss map and commented on it, “The Gauss map has been shown to be a good example of a chaotic discrete dynamical system”; however, “The numerical simulation of the map behaves significantly differently, in that the numerical simulation is not chaotic” (a more detailed discussion is offered in the Appendix).Definition 3.
The third definition of Gauss map is a chaotic series with the definition (Formula 3.42 in Anagnostopoulos [3] and Formula 3.3 in Saha and Das [4]) of (2)
x
n
+
1
=
exp
⁡
-
α
x
n
2
+
β
.This type of definition is also called “mouse map,” since its bifurcation map resembles a mouse (see Figure 1).Figure 1
Bifurcation of Gauss/mouse map of Definition2 (x
(
0
)
=
0.1).
(a)
α = 4.9
(b)
α = 5.2Arasomwan’s Definition. In Section 3.3, formula 9, page 3, Arasomwan and Adewumi [1] stated the following.“3.3. Gaussian.This map is also known as Gauss or mouse map. It is defined as
9
x
k
+
1
=
0
,
x
k
=
0
,
1
x
k
mod
⁡
1
,
x
k
∈
0,1
,
where
(
1
/
x
k
)
mod
⁡
(
1
)
=
(
1
/
x
k
)
-
1
/
x
k
and
z
denotes the largest integer less than
z
which acts as a shift on the continued fraction representation of numbers.”


## 2. Discussion

Arasomwan’s definition of “Gauss map” [1] contains a paradox. From the context (“The map is also known as Gauss or mouse map”), we guess the third definition was appropriate, because this kind of Gauss map has a nickname of “Mouse map.” From formula 9, we guess the second definition was appropriate, because formula 9 was similar to formula (1). It is suggested that the authors either remove the phrase of “Mouse map” or replace formula 9 with formula (2).


## 3. Conclusion

In spite of this issue, [1] is one of the best papers that we have ever read. We enjoyed reading it, which gave many thought-provoking and informative ideas especially in the results that “intermittency map” performed best among all chaotic maps.



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*Source: 815370-2015-04-08.xml*