# A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Chaotic Mappings, Bifurcations, and Lyapunov’s Exponent

#### 2.2. The Chaotic Fish School Search Algorithm with Exponetial Step Decay

## 3. Numerical Experiments

#### 3.1. Performance Comparison of Chaotic and Nonchaotic PRNGs

#### 3.2. Comparison of Different Chaotic Maps Used as PRNGs in FSS with Exponential Step Decay

#### 3.3. Comparison of FSS with Exponential Step Decay with Original FSS, PSO, and GA

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Bifurcation diagrams of the chaotic maps listed in Table 1: (

**a**) Logistic map; (

**b**) Square map; (

**c**) Cosine map; (

**d**) Tent map; (

**e**) Sine map; (

**f**) Circle map.

**Figure 2.**Lyapunov exponent plots of the chaotic maps listed in Table 1: (

**a**) Logistic map; (

**b**) Square map; (

**c**) Cosine map; (

**d**) Tent map; (

**e**) Sine map; (

**f**) Circle map.

**Figure 3.**Histogram comparison of the relative occurrence frequency of random numbers generated by chaotic maps: (

**a**) Logistic map; (

**b**) Square map; (

**c**) Cosine map; (

**d**) Tent map; (

**e**) Sine map; (

**f**) Circle map.

**Figure 5.**Histogram comparison of the relative occurrence frequency of the uniformly distributed pseudorandom numbers generated by (

**a**) Mersenne Twister and (

**b**) the chaotic tent map.

**Figure 6.**Convergence curves of chaotic algorithms based on FSS with exponential step decay, optimizing 15-dimensional versions of the respective test functions listed in Table 4: (

**a**) ${f}_{1}$ (

**b**) ${f}_{2}$ (

**c**) ${f}_{3}$ (

**d**) ${f}_{4}$ (

**e**) ${f}_{5}$ (

**f**) ${f}_{6}$ (

**g**) ${f}_{7}$ (

**h**) ${f}_{8}$.

**Figure 7.**Convergence curves of PSO, FSS, GA, and ETFSS algorithms optimizing the 15-dimensional versions of test functions listed in Table 4: (

**a**) ${f}_{1}$ (

**b**) ${f}_{2}$ (

**c**) ${f}_{3}$ (

**d**) ${f}_{4}$ (

**e**) ${f}_{5}$ (

**f**) ${f}_{6}$ (

**g**) ${f}_{7}$ (

**h**) ${f}_{8}$.

**Figure 8.**Boxplot of the solutions obtained by PSO, FSS, GA, and ETFSS optimizing the 15-dimensional version of the multidimensional Styblinsky-Tang test function.

Name | Equation | $\mathit{\mu}$ |
---|---|---|

Logistic map | ${y}_{n+1}=\mu {y}_{n}\left(1-{y}_{n}\right)$ | $4.0$ |

Square map | ${y}_{n+1}=1-\mu {y}_{n}^{2}$ | $2.0$ |

Cosine map | ${y}_{n+1}=\mathrm{cos}\left(\mu {y}_{n}\right)$ | $6.0$ |

Tent map | ${y}_{n+1}=\mu \text{}\mathrm{min}\left\{{y}_{n},1-{y}_{n}\right\}$ | $1.9999$ |

Sine map | ${y}_{n+1}=-\mu \mathrm{sin}\left(y\right)$ | $4.0$ |

Circle map | ${y}_{n+1}={y}_{n}-\mu \mathrm{sin}\left({y}_{n}\right)$ | $4.5$ |

Parameter | Value |
---|---|

Processor type | Intel^{®} Core™ i7-4770 |

Processor clock rate | 3.40 GHz (4 physical cores) |

L2\L3 processor cache sizes | 1024 KiB\8192 KiB |

Random access memory | 16 GB DDR3 (1600 MHz) |

PRNG | Mean | SD | Best |
---|---|---|---|

Logistic map | $1.693\times {10}^{8}$ | $4.584\times {10}^{4}$ | $1.694\times {10}^{8}$ |

Square map | $1.223\times {10}^{8}$ | $8.826\times {10}^{4}$ | $1.225\times {10}^{8}$ |

Cosine map | $4.020\times {10}^{7}$ | $6.587\times {10}^{3}$ | $4.021\times {10}^{7}$ |

Tent map | $1.685\times {10}^{8}$ | $4.574\times {10}^{4}$ | $1.686\times {10}^{8}$ |

Sine map | $3.924\times {10}^{7}$ | $1.002\times {10}^{3}$ | $3.926\times {10}^{7}$ |

Circle map | $4.511\times {10}^{7}$ | $2.212\times {10}^{3}$ | $4.516\times {10}^{7}$ |

Mersenne Twister | $4.833\times {10}^{7}$ | $9.687\times {10}^{3}$ | $4.835\times {10}^{7}$ |

PCG | $8.552\times {10}^{7}$ | $2.534\times {10}^{4}$ | $8.558\times {10}^{7}$ |

Philox | $2.052\times {10}^{7}$ | $4.360\times {10}^{3}$ | $2.053\times {10}^{7}$ |

**Table 4.**Test functions used to benchmark the chaotic optimization algorithms. M and U letters denote multimodal and unimodal optimization problems respectively.

Test Function Formula | Dim | Region | Optimum | Type |
---|---|---|---|---|

${f}_{1}\left(\overrightarrow{x}\right)=An+{\mathsf{\Sigma}}_{i=1}^{n}\left({x}_{i}^{2}-Acos\left(2\pi {x}_{i}\right)\right),\text{}A=10.$ | 15 | $\left[-5.12,\text{}5.12\right]$ | $0$ | M |

${f}_{2}\left(\overrightarrow{x}\right)={\mathsf{\Sigma}}_{i=1}^{n}\left(\frac{{x}_{i}^{2}}{4000}\right)-{\mathsf{\Pi}}_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{{i}^{0.5}}\right)-1.$ | 15 | $\left[-100,\text{}100\right]$ | $0$ | M |

${f}_{3}\left(\overrightarrow{x}\right)=0.5{\mathsf{\Sigma}}_{i=1}^{n}\left({x}_{i}^{4}-16{x}_{i}^{2}+5{x}_{i}\right).$ | 15 | $\left[-5,\text{}5\right]$ | $-39.1659n$ | M |

${f}_{4}\left(\overrightarrow{x}\right)={\mathsf{\Sigma}}_{i=1}^{n}{\left({\mathsf{\Sigma}}_{j=1}^{i}{x}_{j}\right)}^{2}.$ | 15 | $\left[-100,\text{}100\right]$ | $0$ | U |

${f}_{5}\left(\overrightarrow{x}\right)=-20\mathrm{exp}\left(-0.2{\left(\frac{1}{n}{\mathsf{\Sigma}}_{i=1}^{n}{x}_{i}^{2}\right)}^{0.5}\right)$$-\mathrm{exp}\left(\frac{1}{n}{\mathsf{\Sigma}}_{i=1}^{n}\mathrm{cos}2\pi {x}_{i}\right)+e+20.$ | 15 | $\left[-32.7,\text{}32.7\right]$ | $0$ | M |

${f}_{6}\left(\overrightarrow{x}\right)={\mathsf{\Sigma}}_{i=1}^{n}{x}_{i}^{2}.$ | 15 | $\left[-100,\text{}100\right]$ | $0$ | U |

${f}_{7}\left(\overrightarrow{x}\right)={\mathsf{\Sigma}}_{i=1}^{n-1}\left(100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right).$ | 15 | $\left[-10,\text{}10\right]$ | $0$ | U |

${f}_{8}\left(\overrightarrow{x}\right)={\mathsf{\Sigma}}_{i=1}^{n}{x}_{i}^{2}+{\left(\frac{1}{2}{\mathsf{\Sigma}}_{i=1}^{n}i{x}_{i}\right)}^{2}+{\left(\frac{1}{2}{\mathsf{\Sigma}}_{i=1}^{n}i{x}_{i}\right)}^{4}.$ | 15 | $\left[-10,\text{}10\right]$ | $0$ | U |

**Table 5.**Accuracy comparison of the considered algorithms based on fish school search (FSS) with exponential step decay (EFSS). Algorithms based on the logistic, square, cosine, tent, sine, and circle maps are abbreviated as ELFSS, ESFSS, ECFSS, ETFSS, ESiFSS, and ECiFSS respectively. The best result in each row is highlighted in bold.

Function | Metric | EFSS | ELFSS | ESFSS | ECFSS | ESiFSS | ECiFSS | ETFSS |
---|---|---|---|---|---|---|---|---|

${f}_{1}$ | Best | $-3.00$ | $-3.00$ | $-3.00$ | $-57.73$ | $-3.00$ | $\mathbf{-}\mathbf{2.00}$ | $-2.99$ |

Mean | $\mathbf{-}\mathbf{5.98}$ | $-23.07$ | $-6.78$ | $-92.76$ | $\mathbf{-}\mathbf{5.82}$ | $-6.75$ | $-6.48$ | |

SD | $\mathbf{1.99}$ | $13.67$ | $2.06$ | $19.81$ | $2.31$ | $3.13$ | $2.37$ | |

${f}_{2}$ | Best | $-0.001$ | $-0.003$ | $-0.002$ | $-0.003$ | $-0.002$ | $-0.001$ | $\mathbf{-}\mathbf{0.001}$ |

Mean | $-0.06$ | $-1.53$ | $-0.01$ | $-0.01$ | $-0.01$ | $\mathbf{-}\mathbf{0.00}$ | $-0.03$ | |

SD | $0.206$ | $\text{}1.278$ | $0.006$ | $0.006$ | $0.008$ | $\mathbf{0.005}$ | $0.162$ | |

${f}_{3}$ | Best | $559.22$ | $530.94$ | $\mathbf{587.49}$ | $418.46$ | $559.22$ | $545.08$ | $573.36$ |

Mean | $504.21$ | $\text{}428.68$ | $485.08$ | $387.27$ | $500.57$ | $504.757$ | $\mathbf{509.89}$ | |

SD | $27.80$ | $69.77$ | $48.48$ | $\mathbf{13.96}$ | $\text{}38.19$ | $\text{}33.34$ | $35.15$ | |

${f}_{4}$ | Best | $\mathbf{-}\mathbf{0.03}$ | $-0.11$ | $-0.13$ | $-0.17$ | $-0.10$ | $-0.11$ | $-0.04$ |

Mean | $-0.10$ | $-8\times {10}^{4}$ | $-0.18$ | $-3.36$ | $-0.16$ | $-0.15$ | $\mathbf{-}\mathbf{0.10}$ | |

SD | $0.03$ | $7\times {10}^{4}$ | $0.02$ | $6.31$ | $0.03$ | $\text{}0.02$ | $\mathbf{0.02}$ | |

${f}_{5}$ | Best | $\mathbf{-}\mathbf{0.03}$ | $-0.06$ | $-0.05$ | $-0.07$ | $-0.05$ | $-0.05$ | $-0.04$ |

Mean | $-0.05$ | $-9.84$ | $-0.07$ | $-6.74$ | $-0.06$ | $-0.06$ | $\mathbf{-}\mathbf{0.05}$ | |

SD | $0.006$ | $8.333$ | $0.008$ | $4.597$ | $0.007$ | $\mathbf{0.005}$ | $0.006$ | |

${f}_{6}$ | Best | $-0.01$ | $-0.01$ | $-0.02$ | $-0.02$ | $-0.02$ | $-0.02$ | $\mathbf{-}\mathbf{0.01}$ |

Mean | $-0.019$ | $-5\times {10}^{3}$ | $-0.031$ | $-0.035$ | $-0.026$ | $-0.025$ | $\mathbf{-}\mathbf{0.018}$ | |

SD | $\mathbf{0.003}$ | $5\times {10}^{3}$ | $0.006$ | $0.005$ | $0.005$ | $0.003$ | $0.003$ | |

${f}_{7}$ | Best | $-8.73$ | $-9.00$ | $-8.60$ | $\mathbf{-}\mathbf{5.76}$ | $-8.63$ | $-8.21$ | $-8.36$ |

Mean | $-10.15$ | $-5\times {10}^{5}$ | $-13.13$ | $\mathbf{-}\mathbf{5.98}$ | $-9.82$ | $-9.96$ | $-9.81$ | |

SD | $1.29$ | $4\times {10}^{5}$ | $15.52$ | $\mathbf{0.12}$ | $0.71$ | $0.84$ | $0.80$ | |

${f}_{8}$ | Best | $-0.00$ | $-0.00$ | $-0.00$ | $-42.83$ | $-0.00$ | $-0.00$ | $\mathbf{-}\mathbf{0.00}$ |

Mean | $\mathbf{-}\mathbf{1.64}$ | $-74.44$ | $-15.93$ | $-91.28$ | $-6.20$ | $-7.09$ | $-3.70$ | |

SD | $\mathbf{6.26}$ | $60.87$ | $42.04$ | $35.18$ | $17.13$ | $19.29$ | $9.67$ |

**Table 6.**Accuracy comparison of PSO, GA with tournament selection, FSS, and ETFSS on 15-dimensional test functions. The best result in each row is highlighted in bold.

Function | Metric | PSO | FSS | GA | ETFSS |
---|---|---|---|---|---|

${f}_{1}$ | Best | $-17.956$ | $-1.996$ | $-9.952$ | $\mathbf{-}\mathbf{1.998}$ |

Mean | $-32.677$ | $-9.585$ | $-23.963$ | $\mathbf{-}\mathbf{5.448}$ | |

SD | $11.266$ | $7.138$ | $8.903$ | $\mathbf{1.901}$ | |

${f}_{2}$ | Best | $-0.028$ | $-0.011$ | $-0.05$ | $\mathbf{-}\mathbf{0.001}$ |

Mean | $-0.098$ | $-0.835$ | $-0.201$ | $\mathbf{-}\mathbf{0.008}$ | |

SD | $0.101$ | $0.162$ | $0.093$ | $\mathbf{0.010}$ | |

${f}_{3}$ | Best | $545.080$ | $587.459$ | $573.689$ | $\mathbf{587.472}$ |

Mean | $508.096$ | $516.325$ | $\mathbf{568.118}$ | $528.118$ | |

SD | $18.570$ | $46.613$ | $\mathbf{4.523}$ | $48.791$ | |

${f}_{4}$ | Best | $\mathbf{-}\mathbf{0.007}$ | $-0.050$ | $-4.675$ | $-0.045$ |

Mean | $-11.900$ | $-31.630$ | $-9.056$ | $\mathbf{-}\mathbf{5.457}$ | |

SD | $25.600$ | $64.923$ | $\mathbf{2.482}$ | $28.862$ | |

${f}_{5}$ | Best | $-1.647$ | $-0.038$ | $-1.383$ | $\mathbf{-}\mathbf{0.037}$ |

Mean | $-3.074$ | $-1.726$ | $-2.185$ | $\mathbf{-}\mathbf{0.052}$ | |

SD | $0.899$ | $1.587$ | $0.449$ | $\mathbf{0.005}$ | |

${f}_{6}$ | Best | $-0.001$ | $\mathbf{-}\mathbf{0.008}$ | $-0.697$ | $-0.012$ |

Mean | $-1.589$ | $-6.345$ | $-1.226$ | $\mathbf{-}\mathbf{0.018}$ | |

SD | $2.317$ | $14.178$ | $0.264$ | $\mathbf{0.003}$ | |

${f}_{7}$ | Best | $-13.074$ | $-10.018$ | $-93.094$ | $\mathbf{-}\mathbf{7.886}$ |

Mean | $-127.166$ | $-24.882$ | $-293.899$ | $\mathbf{-}\mathbf{9.749}$ | |

SD | $184.037$ | $20.196$ | $191.647$ | $\mathbf{0.766}$ | |

${f}_{8}$ | Best | $-0.037$ | $-0.001$ | $-74.492$ | $\mathbf{-}\mathbf{0.000}$ |

Mean | $\mathbf{-}\mathbf{2.278}$ | $-13.895$ | $-228.221$ | $-4.238$ | |

SD | $\mathbf{2.284}$ | $26.085$ | $82.664$ | $10.912$ |

Function | PSO | FSS | GA |
---|---|---|---|

${f}_{1}$ | $+$ | $+$ | $+$ |

${f}_{2}$ | $+$ | $+$ | $+$ |

${f}_{3}$ | $+$ | $+$ | $-$ |

${f}_{4}$ | $+$ | $=$ | $+$ |

${f}_{5}$ | $+$ | $+$ | $+$ |

${f}_{6}$ | $+$ | $=$ | $+$ |

${f}_{7}$ | $+$ | $+$ | $+$ |

${f}_{8}$ | $=$ | $=$ | $+$ |

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Demidova, L.A.; Gorchakov, A.V.
A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay. *Symmetry* **2020**, *12*, 784.
https://doi.org/10.3390/sym12050784

**AMA Style**

Demidova LA, Gorchakov AV.
A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay. *Symmetry*. 2020; 12(5):784.
https://doi.org/10.3390/sym12050784

**Chicago/Turabian Style**

Demidova, Liliya A., and Artyom V. Gorchakov.
2020. "A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay" *Symmetry* 12, no. 5: 784.
https://doi.org/10.3390/sym12050784