Research on Information Identification of Chaotic Map with MultiStability
Abstract
:1. Introduction
2. Information Identification of Chaotic Map
3. The Proposed Intelligent Optimization Algorithm
3.1. Classical Sparrow Search Algorithm
3.2. Improved Sparrow Search Algorithm
Algorithm 1 Pseudo code of ISSA 

4. Identification of Initial Values and System Parameters
4.1. Identification Systems
4.2. Numerical Simulation
4.3. Results Analysis
5. Identification with Any Initial Values
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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System  System Parameter  Initial Value 

QDM  $k=1.78$  ${x}_{0}=0.5,{y}_{0}=0.5$ 
ADM  $k=2.3$  ${x}_{0}=0.8,{y}_{0}=0.6$ 
SDM  $k=1.84$  ${x}_{0}=0.5,{y}_{0}=0.6$ 
Algorithm  Parameter Setting 

DE  $CR=F=0.6$ 
PSO  ${c}_{1}={c}_{2}=2,{\omega}_{min}=0.4,{\omega}_{max}=0.9$ 
ABC  $limit=100$ 
BSA  $C=S=1.5,{a}_{1}={a}_{2}=1,FQ=3,P\in [0.8,1],FL\in [0.5,0.9]$ 
JAYA   
SSA  $ST=0.8$ 
ISSA  $ST=0.8,a=0.6$ 
Algorithm  Min J  Average J  Standard Deviation  Accurate Identification 

DE  1.0596 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{9}$  7.6312 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$  1.1302 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$ 
$k=1.7800,{x}_{0}=0.5000,{y}_{0}=0.5000$ $k=1.7800,{x}_{0}=0.6514,{y}_{0}=0.6514$ 
PSO  3.6528 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{14}$  6.1925 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{11}$  1.6995 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{10}$ 
$k=1.7800,{x}_{0}=0.5000,{y}_{0}=0.5000$ $k=1.7800,{x}_{0}=0.6514,{y}_{0}=0.6514$ $k=1.7800,{x}_{0}=1.1514,{y}_{0}=1.1514$ 
ABC  1.2452 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  1.3061 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  1.2951 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$   
BSA  7.1560 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{14}$  1.8623 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  1.1445 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$ 
$k=1.7800,{x}_{0}=0.5000,{y}_{0}=0.5000$ $k=1.7800,{x}_{0}=0.6514,{y}_{0}=0.6514$ $k=1.7800,{x}_{0}=1.1514,{y}_{0}=1.1514$ 
JAYA  7.2978 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  9.1129 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  8.5444 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$   
SSA  3.3176 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{15}$  1.1539 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  8.1394 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$ 
$k=1.7800,{x}_{0}=0.5000,{y}_{0}=0.5000$ $k=1.7800,{x}_{0}=0.6514,{y}_{0}=0.6514$ $k=1.7800,{x}_{0}=1.1514,{y}_{0}=1.1514$ 
ISSA  8.2173 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{32}$  2.7615 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{28}$  1.5450 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{27}$ 
$k=1.7800,{x}_{0}=0.5000,{y}_{0}=0.5000$ $k=1.7800,{x}_{0}=0.6514,{y}_{0}=0.6514$ $k=1.7800,{x}_{0}=1.1514,{y}_{0}=1.1514$ 
Algorithm  Min J  Average J  Standard Deviation  Accurate Identification 

DE  2.5280 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{12}$  1.5302 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$  5.4263 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$ 
$k=2.3000,{x}_{0}=0.8000,{y}_{0}=0.6000$ $k=2.3000,{x}_{0}=0.4000,{y}_{0}=0.2000$ 
PSO  6.8340 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{14}$  9.0310 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{11}$  3.0218 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{10}$ 
$k=2.3000,{x}_{0}=0.8000,{y}_{0}=0.6000$ $k=2.3000,{x}_{0}=0.4000,{y}_{0}=0.2000$ $k=2.3000,{x}_{0}=1.0928,{y}_{0}=1.2928$ 
ABC  1.0000 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{8}$  2.1014 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  4.2926 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  $k=2.3000,{x}_{0}=0.4000,{y}_{0}=0.2000$ 
BSA  1.7174 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{14}$  3.7102 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  2.5935 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$ 
$k=2.3000,{x}_{0}=0.8000,{y}_{0}=0.6000$ $k=2.3000,{x}_{0}=0.4000,{y}_{0}=0.2000$ $k=2.3000,{x}_{0}=1.0928,{y}_{0}=1.2928$ 
JAYA  3.1510 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  3.6840 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  4.5653 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$   
SSA  5.7673 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{22}$  6.4127 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{13}$  4.5076 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{12}$ 
$k=2.3000,{x}_{0}=0.8000,{y}_{0}=0.6000$ $k=2.3000,{x}_{0}=0.4000,{y}_{0}=0.2000$ $k=2.3000,{x}_{0}=1.0928,{y}_{0}=1.2928$ 
ISSA  3.2689 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{31}$  3.2823 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{29}$  8.4737 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{29}$ 
$k=2.3000,{x}_{0}=0.8000,{y}_{0}=0.6000$ $k=2.3000,{x}_{0}=0.4000,{y}_{0}=0.2000$ $k=2.3000,{x}_{0}=1.0928,{y}_{0}=1.2928$ 
Algorithm  Min J  Average J  Standard Deviation  Accurate Identification 

DE  1.2100 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  1.4477 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  2.1841 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$   
PSO  2.7861 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{11}$  4.0176 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  8.1275 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$ 
$k=1.8400,{x}_{0}=0.5000,{y}_{0}=0.4000$ $k=1.8400,{x}_{0}=0.9289,{y}_{0}=0.8289$ $k=1.8400,{x}_{0}=0.6245,{y}_{0}=0.7245$ 
ABC  4.8297 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  4.9959 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  7.6374 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$   
BSA  1.7721 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{12}$  6.3962 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  1.1652 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  $k=1.8400,{x}_{0}=0.6245,{y}_{0}=0.7245$ 
JAYA  2.6764 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  1.7661 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$  2.2592 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$   
SSA  1.0565 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{8}$  1.4487 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  5.8847 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{2}$   
ISSA  7.0006 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{26}$  8.3717 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  1.2630 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$ 
$k=1.8400,{x}_{0}=0.5000,{y}_{0}=0.4000$ $k=1.8400,{x}_{0}=0.9289,{y}_{0}=0.8289$ $k=1.8400,{x}_{0}=0.6245,{y}_{0}=0.7245$ 
Algorithm  Min J  Average J  Standard Deviation  Accurate Identification 

DE  4.4276 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{9}$  1.2313 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  2.6389 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  $k=1.5000,{x}_{0}=0.1000,{y}_{0}=0.1000$ 
PSO  8.6821 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{11}$  5.1173 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{9}$  1.5569 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{9}$  $k=1.5000,{x}_{0}=0.1000,{y}_{0}=0.1000$ 
ABC  9.4028 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  1.2273 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  1.2005 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$   
BSA  2.9036 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{13}$  2.4014 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  9.0801 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  $k=1.5000,{x}_{0}=0.1000,{y}_{0}=0.1000$ 
JAYA  5.8166 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$  6.4159 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  2.6890 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$   
SSA  5.5330 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{12}$  6.3569 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  1.8484 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  $k=1.5000,{x}_{0}=0.1000,{y}_{0}=0.1000$ 
ISSA  3.0417 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{30}$  3.9779 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{11}$  2.0097 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{10}$  $k=1.5000,{x}_{0}=0.1000,{y}_{0}=0.1000$ 
System  Min J  Average J  Standard Deviation  Accurate Identification 

QDM  1.1620 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{31}$  1.1620 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{31}$  2.2270 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{47}$  $k=1.780000000000000$ 
ADM  4.1087 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{32}$  4.1087 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{32}$  1.1135 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{47}$  $k=2.300000000000000$ 
SDM  8.1403 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{32}$  8.1403 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{32}$  2.2270 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{47}$  $k=1.840000000000001$ 
System  SNR(dB)  Average J  Average Identification Result  RE 

QDM  10  8.5678 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  0.48101725418938  73.98% 
20  9.1598 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  1.63532335966099  11.12%  
30  8.0724 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  1.77856372619514  3.34%  
40  8.5505 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  1.77873077715485  0.07%  
50  7.9381 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$  1.77958662235126  0.02%  
ADM  10  7.6060 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  2.2511417157987  2.12% 
20  8.4544 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  2.28961739751150  0.45%  
30  8.3162 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  2.30163725145009  0.07%  
40  7.5631 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  2.30019465294567  0.01%  
50  7.4071 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$  2.30012024008413  0.01%  
SDM  10  9.3754 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$  1.3665566983129  25.73% 
20  7.6273 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{4}$  1.82645413905643  0.74%  
30  7.4214 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$  1.83699837846524  0.30%  
40  7.9969 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$  1.83945874768613  0.05%  
50  9.1389 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$  1.84042638568128  0.02% 
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Li, Y.; Peng, Y. Research on Information Identification of Chaotic Map with MultiStability. Fractal Fract. 2023, 7, 811. https://doi.org/10.3390/fractalfract7110811
Li Y, Peng Y. Research on Information Identification of Chaotic Map with MultiStability. Fractal and Fractional. 2023; 7(11):811. https://doi.org/10.3390/fractalfract7110811
Chicago/Turabian StyleLi, You, and Yuexi Peng. 2023. "Research on Information Identification of Chaotic Map with MultiStability" Fractal and Fractional 7, no. 11: 811. https://doi.org/10.3390/fractalfract7110811