Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 815370, 3 pages

http://dx.doi.org/10.1155/2015/815370

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

^{1}School of Computer Science and Technology, Nanjing Normal University, Nanjing, Jiangsu 210023, China^{2}Translational Imaging Division and MRI Unit, Columbia University and New York State Psychiatric Institute, New York, NY 10032, USA^{3}School of Natural Sciences and Mathematics, Shepherd University, Shepherdstown, WV 25443, USA

Received 9 February 2015; Accepted 23 March 2015

Academic Editor: Marzio Pennisi

Copyright © 2015 Yudong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 space to the unit spheres ; that is, given a surface lying in , the Gauss map is a continuous map such that is a unit vector orthogonal to at , namely, the normal vector to at .

*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 iswhere 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 This type of definition is also called “mouse map,” since its bifurcation map resembles a mouse (see Figure 1).