Skip to main content
Springer Nature Link
Log in

Fractional-order chaotic oscillator-based Aquila optimization algorithm for maximization of the chaotic with Lorentz oscillator

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

A Correction to this article was published on 08 November 2023

This article has been updated

Abstract

The random structures in the Aquila optimization algorithm are modeled with fractional chaotic oscillators, and the fractional-order chaotic oscillator-based Aquila optimization (FOCOBAO) algorithm was suggested in this study. First of all, the basic AO algorithm was examined. In particular, random variables that affect the optimization performance of the AO algorithm have been determined. Then, instead of the determined random variables, the coefficients were derived with fractional chaotic oscillators and used in the FOCOBAO. The superiority of the proposed algorithm was primarily demonstrated via twenty-three benchmark functions. The results were matched with GO, EO, GWO, MPA, WOA, SMA and basic AO optimization algorithms. Then, the design of the Lorenz chaotic oscillator, according to maximum chaotic objective function, is a topic that remains up to date in the literature. In this study, a fractional chaotic Lorenz oscillator was designed with FOCOBAO as an engineering application. Especially for maximum chaoticity, maximum positive Lyapunov exponents were determined. In this way, a different design process has been proposed in the literature. The basic AO algorithm, which includes stochastic processes, was developed with fractional chaotic oscillators, and a deterministic method was obtained. The parameters of the Lorenz system were calculated for maximum chaoticity, and the results were presented comparatively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Data availability

Data will be made available on reasonable request.

Change history

References

  1. Coello CAC (2022) Constraint-handling techniques used with evolutionary algorithms. In: GECCO 2022 Companion—Proc 2022 Genet Evol Comput Conf pp 1310–1333. https://doi.org/10.1145/3520304.3533640

  2. Folino G, Forestiero A, Spezzano G (2006) A jxta based asynchronous peer-to-peer implementation of genetic programming. J Softw 1:12–23. https://doi.org/10.4304/jsw.1.2.12-23

    Article  Google Scholar 

  3. Forestiero A (2017) Bio-inspired algorithm for outliers detection. Multimed Tools Appl 76:25659–25677. https://doi.org/10.1007/s11042-017-4443-1

    Article  Google Scholar 

  4. Abualigah L, Elaziz MA, Khasawneh AM et al (2022) Meta-heuristic optimization algorithms for solving real-world mechanical engineering design problems: a comprehensive survey, applications, comparative analysis, and results. Neural Comput Appl 346(34):4081–4110. https://doi.org/10.1007/S00521-021-06747-4

    Article  Google Scholar 

  5. Storn R, Price K (1997) Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11:341–359

    Article  MathSciNet  MATH  Google Scholar 

  6. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput doi 10(1109/4235):771163

    Google Scholar 

  7. Grefenstette JJ Genetic algorithms and machine learning

  8. Gandomi AH (2014) Interior search algorithm (ISA): a novel approach for global optimization. ISA Trans 53:1168–1183. https://doi.org/10.1016/J.ISATRA.2014.03.018

    Article  Google Scholar 

  9. Koza JR (1994) Genetic programming as a means for programming computers by natural selection. Stat Comput 4:87–112. https://doi.org/10.1007/BF00175355/METRICS

    Article  Google Scholar 

  10. Abualigah L, Diabat A, Mirjalili S et al (2021) The arithmetic optimization algorithm. Comput Methods Appl Mech Eng 376:113609. https://doi.org/10.1016/J.CMA.2020.113609

    Article  MathSciNet  MATH  Google Scholar 

  11. Moghdani R, Salimifard K (2018) Volleyball premier league algorithm. Appl Soft Comput J 64:161–185. https://doi.org/10.1016/j.asoc.2017.11.043

    Article  Google Scholar 

  12. Dai C, Zhu Y, Chen W (2007) Seeker optimization algorithm. pp 167–176

  13. Rao RV, Savsani VJ, Vakharia DP (2011) Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Des 43:303–315. https://doi.org/10.1016/J.CAD.2010.12.015

    Article  Google Scholar 

  14. Mousavirad SJ, Ebrahimpour-Komleh H (2017) Human mental search: a new population-based metaheuristic optimization algorithm. Appl Intell 47:850–887. https://doi.org/10.1007/S10489-017-0903-6/TABLES/17

    Article  Google Scholar 

  15. Samuel P, Subbaiyan S, Balusamy B et al (2021) A technical survey on intelligent optimization grouping algorithms for finite state automata in deep packet inspection. Arch Comput Methods Eng 28:1371–1396. https://doi.org/10.1007/S11831-020-09419-Z/FIGURES/5

    Article  MathSciNet  Google Scholar 

  16. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007

    Article  Google Scholar 

  17. Alatas B (2011) ACROA: artificial chemical reaction optimization algorithm for global optimization. Expert Syst Appl 38:13170–13180. https://doi.org/10.1016/J.ESWA.2011.04.126

    Article  Google Scholar 

  18. Zhao W, Wang L, Zhang Z (2019) Atom search optimization and its application to solve a hydrogeologic parameter estimation problem. Knowledge-Based Syst 163:283–304. https://doi.org/10.1016/J.KNOSYS.2018.08.030

    Article  Google Scholar 

  19. Nematollahi AF, Rahiminejad A, Vahidi B (2017) A novel physical based meta-heuristic optimization method known as lightning attachment procedure optimization. Appl Soft Comput 59:596–621. https://doi.org/10.1016/J.ASOC.2017.06.033

    Article  Google Scholar 

  20. Faramarzi A, Heidarinejad M, Stephens B, Mirjalili S (2020) Equilibrium optimizer: a novel optimization algorithm. Knowledge-Based Syst. https://doi.org/10.1016/j.knosys.2019.105190

    Article  Google Scholar 

  21. Xie L, Zeng J, Cui Z (2009) General framework of artificial physics optimization algorithm. In: Proceedings 2009 World congress on nature and biologically inspired computing, NABIC

  22. Sharma M, Kaur P (2021) A comprehensive analysis of nature-inspired meta-heuristic techniques for feature selection problem. Arch Comput Methods Eng 28:1103–1127. https://doi.org/10.1007/S11831-020-09412-6/TABLES/17

    Article  MathSciNet  Google Scholar 

  23. Mirjalili S, Gandomi AH, Mirjalili SZ et al (2017) Salp swarm algorithm: a bio-inspired optimizer for engineering design problems. Adv Eng Softw. https://doi.org/10.1016/j.advengsoft.2017.07.002

    Article  Google Scholar 

  24. Gandomi AH, Alavi AH (2012) Krill herd: a new bio-inspired optimization algorithm. Commun Nonlinear Sci Numer Simul 17:4831–4845. https://doi.org/10.1016/J.CNSNS.2012.05.010

    Article  MathSciNet  MATH  Google Scholar 

  25. Jain M, Singh V, Rani A (2019) A novel nature-inspired algorithm for optimization: squirrel search algorithm. Swarm Evol Comput. https://doi.org/10.1016/j.swevo.2018.02.013

    Article  Google Scholar 

  26. Mirjalili S, SMAL, (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61

    Article  Google Scholar 

  27. Eberhart R, Kennedy J (1995) New optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science pp 39–43. https://doi.org/10.1109/MHS.1995.494215

  28. Kashani AR, Chiong R, Mirjalili S, Gandomi AH (2021) Particle swarm optimization variants for solving geotechnical problems: review and comparative analysis. Arch Comput Methods Eng 28:1871–1927. https://doi.org/10.1007/S11831-020-09442-0/TABLES/16

    Article  MathSciNet  Google Scholar 

  29. Eskandar H, Sadollah A, Bahreininejad A, Hamdi M (2012) Water cycle algorithm—a novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput Struct 110–111:151–166. https://doi.org/10.1016/J.COMPSTRUC.2012.07.010

    Article  Google Scholar 

  30. Bayraktar Z, Komurcu M, Werner DH (2010) Wind driven optimization (WDO): a novel nature-inspired optimization algorithm and its application to electromagnetics. In: 2010 IEEE Int Symp Antennas Propag CNC-USNC/URSI Radio Sci Meet - Lead Wave, AP-S/URSI 2010. https://doi.org/10.1109/APS.2010.5562213

  31. Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680. https://doi.org/10.1126/SCIENCE.220.4598.671

    Article  MathSciNet  MATH  Google Scholar 

  32. Ewees AA, Abualigah L, Yousri D et al (2022) Improved slime mould algorithm based on firefly algorithm for feature selection: a case study on QSAR model. Eng Comput 38:2407–2421. https://doi.org/10.1007/S00366-021-01342-6/TABLES/13

    Article  Google Scholar 

  33. Şahin CB, Dinler ÖB, Abualigah L (2021) Prediction of software vulnerability based deep symbiotic genetic algorithms: phenotyping of dominant-features. Appl Intell 51:8271–8287. https://doi.org/10.1007/S10489-021-02324-3/FIGURES/7

    Article  Google Scholar 

  34. Forestiero A (2016) Self-organizing anomaly detection in data streams. Inf Sci 373:321–336

    Article  Google Scholar 

  35. Wang WC, Xu L, Chau KW, Zhao Y, Xu DM et al (2022) An orthogonal opposition-based-learning Yin–Yang-pair optimization algorithm for engineering optimization. Eng Comput 38:1149–1183. https://doi.org/10.1007/S00366-020-01248-9/FIGURES/10

    Article  Google Scholar 

  36. Kundu T, Garg H (2022) A hybrid ITLHHO algorithm for numerical and engineering optimization problems. Int J Intell Syst 37:3900–3980. https://doi.org/10.1002/INT.22707

    Article  Google Scholar 

  37. Ferreira MP, Rocha ML, Silva Neto AJ, Sacco WF (2018) A constrained ITGO heuristic applied to engineering optimization. Expert Syst Appl 110:106–124. https://doi.org/10.1016/J.ESWA.2018.05.027

    Article  Google Scholar 

  38. Rao H, Jia H, Wu D et al (2022) A modified group teaching optimization algorithm for solving constrained engineering optimization problems. Math 10:3765. https://doi.org/10.3390/MATH10203765

    Article  Google Scholar 

  39. Han X, Xu Q, Yue L et al (2020) An improved crow search algorithm based on spiral search mechanism for solving numerical and engineering optimization problems. IEEE Access 8:92363–92382. https://doi.org/10.1109/ACCESS.2020.2980300

    Article  Google Scholar 

  40. Truby RL, Della SC, Rus D (2020) Distributed proprioception of 3d configuration in soft, sensorized robots via deep learning. IEEE Robot Autom Lett 5:3299–3306. https://doi.org/10.1109/LRA.2020.2976320

    Article  Google Scholar 

  41. Huang Q, Huang R, Hao W et al (2020) Adaptive power system emergency control using deep reinforcement learning. IEEE Trans Smart Grid 11:1171–1182. https://doi.org/10.1109/TSG.2019.2933191

    Article  Google Scholar 

  42. Stokes JM, Yang K, Swanson K et al (2020) A deep learning approach to antibiotic discovery. Cell 180:688-702e13. https://doi.org/10.1016/j.cell.2020.01.021

    Article  Google Scholar 

  43. Banan A, Nasiri A, Taheri-Garavand A (2020) Deep learning-based appearance features extraction for automated carp species identification. Aquac Eng 89:102053. https://doi.org/10.1016/j.aquaeng.2020.102053

    Article  Google Scholar 

  44. Ning C, You F (2019) Optimization under uncertainty in the era of big data and deep learning: when machine learning meets mathematical programming. Comput Chem Eng 125:434–448. https://doi.org/10.1016/j.compchemeng.2019.03.034

    Article  Google Scholar 

  45. Petráš I (2011) Fractional-Order Nonlinear Systems

  46. Ates A (2021) Enhanced equilibrium optimization method with fractional order chaotic and application engineering. Neural Comput Appl 33(16):9849–9876. https://doi.org/10.1007/s00521-021-05756-7

    Article  Google Scholar 

  47. Zainel QM, Darwish SM, Khorsheed MB (2022) Employing quantum fruit fly optimization algorithm for solving three-dimensional chaotic equations. Math 10:4147. https://doi.org/10.3390/MATH10214147

    Article  Google Scholar 

  48. Valencia-Ponce MA, Tlelo-Cuautle E, De La Fraga LG (2021) Estimating the highest time-step in numerical methods to enhance the optimization of chaotic oscillators. Math 9:1938. https://doi.org/10.3390/MATH9161938

    Article  Google Scholar 

  49. Ates A, Chen YQ (2021) Fractional order chaotic model based enhanced equilibrium optimization algorithm for controller design of 3 DOF hover flight system. Proc ASME Des Eng Tech Conf 7:69307. https://doi.org/10.1115/DETC2021-69307

    Article  Google Scholar 

  50. Abualigah L, Yousri D, Abd Elaziz M et al (2021) Aquila optimizer: a novel meta-heuristic optimization algorithm. Comput Ind Eng 157:107250. https://doi.org/10.1016/J.CIE.2021.107250

    Article  Google Scholar 

  51. Abualigah L, Houssein EH, Abd Elaziz M, Oliva D, (2022) Integrating meta-heuristics and machine learning for real-world optimization problems. In: Aquila optimizer based PSO swarm intelligence for IoT task scheduling application in cloud computing. pp 481–497

  52. Gul F, Mir A, Mir I, Mir S (2022) A centralized strategy for multi-agent exploration. IEEE Access 10:126871–126884

    Article  Google Scholar 

  53. Danca MF (2021) Matlab code for lyapunov exponents of fractional-order systems Part II: The Noncommensurate Case. Int J Bifurc Chaos 31:2150187. https://doi.org/10.1142/S021812742150187X

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Engineering Faculty Computer Engineering Department, Inonu University, Malatya, Turkey

    Yakup Cavlak & Abdullah Ateş

  2. Computer Science Department, Prince Hussein Bin Abdullah Faculty for Information Technology, Al Al-Bayt University, Mafraq, 25113, Jordan

    Laith Abualigah

  3. College of Engineering, Yuan Ze University, Taoyuan, Taiwan

    Laith Abualigah

  4. Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, 19328, Jordan

    Laith Abualigah

  5. MEU Research Unit, Middle East University, Amman, Jordan

    Laith Abualigah

  6. Applied Science Research Center, Applied Science Private University, Amman, 11931, Jordan

    Laith Abualigah

  7. School of Computer Sciences, Universiti Sains Malaysia, 11800, Gelugor, Pulau Pinang, Malaysia

    Laith Abualigah

  8. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

    Mohammed Abd Elaziz

  9. Faculty of Computer Science and Engineering, Galala University, Suez, 435611, Egypt

    Mohammed Abd Elaziz

  10. Artificial Intelligence Research Center (AIRC), College of Engineering and Information Technology, Ajman University, Ajman, UAE

    Mohammed Abd Elaziz

  11. Department of Electrical and Computer Engineering, Lebanese American University, Byblos, 13-5053, Lebanon

    Laith Abualigah & Mohammed Abd Elaziz

Authors
  1. Yakup Cavlak
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Abdullah Ateş
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Laith Abualigah
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Mohammed Abd Elaziz
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Abdullah Ateş.

Ethics declarations

Conflict of interest

Authors declare that we have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original online version of this article was revised: The affiliation 11 has been added to author Laith Abualigah.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cavlak, Y., Ateş, A., Abualigah, L. et al. Fractional-order chaotic oscillator-based Aquila optimization algorithm for maximization of the chaotic with Lorentz oscillator. Neural Comput & Applic 35, 21645–21662 (2023). https://doi.org/10.1007/s00521-023-08945-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-023-08945-8

Keywords