Research Papers

Identification of Uncertain Incommensurate Fractional-Order Chaotic Systems Using an Improved Quantum-Behaved Particle Swarm Optimization Algorithm

[+] Author and Article Information
Jiamin Wei

Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China
e-mail: 16118414@bjtu.edu.cn

Yongguang Yu

Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China
e-mail: ygyu@bjtu.edu.cn

Di Cai

Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China
e-mail: 16121602@bjtu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 14, 2017; final manuscript received March 6, 2018; published online March 28, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(5), 051004 (Mar 28, 2018) (12 pages) Paper No: CND-17-1500; doi: 10.1115/1.4039582 History: Received November 14, 2017; Revised March 06, 2018

This paper is concerned with a significant issue in the research of nonlinear science, i.e., parameter identification of uncertain incommensurate fractional-order chaotic systems, which can be essentially formulated as a multidimensional optimization problem. Motivated by the basic particle swarm optimization and quantum mechanics theories, an improved quantum-behaved particle swarm optimization (IQPSO) algorithm is proposed to tackle this complex optimization problem. In this work, both systematic parameters and fractional derivative orders are regarded as independent unknown parameters to be identified. Numerical simulations are conducted to identify two typical incommensurate fractional-order chaotic systems. Simulation results and comparisons analyses demonstrate that the proposed method is suitable for parameter identification with advantages of high effectiveness and efficiency. Moreover, we also, respectively, investigate the effect of systematic parameters, fractional derivative orders, and additional noise on the optimization performances. The corresponding results further validate the superior searching capabilities of the proposed algorithm.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA, p. E2. [PubMed] [PubMed]
Heaviside, O. , 1971, Electromagnetic Theory, Ams Chelsea Publishing, New York.
Ichise, M. , Nagayanagi, Y. , and Kojima, T. , 1971, “ An Analog Simulation of Non-Integer Order Transfer Functions for Analysis of Electrode Processes,” J. Electroanal. Chem. Interfacial Electrochem., 33(2), pp. 253–265. [CrossRef]
Pinto, C. , 2017, “ A Note on Fractional Feed-Forward Networks,” Math. Methods Appl. Sci., 40(17), pp. 6133–6137. [CrossRef]
Pinto, C. M. A. , and Carvalho, A. R. M. , 2017, “ The Impact of Pre-Exposure Prophylaxis (PrEP) and Screening on the Dynamics of HIV,” J. Comput. Appl. Math., in press.
Pinto, C. M. A. , and Carvalho, A. R. M. , 2017, “ The HIV/TB Coinfection Severity in the Presence of TB Multi-Drug Resistant Strains,” Ecol. Complexity, 32(Pt. A), pp. 1–20. [CrossRef]
Pinto, C. M. A. , and Carvalho, A. R. M. , 2017, “ The Role of Synaptic Transmission in a HIV Model With Memory,” Appl. Math. Comput., 292(▪), pp. 76–95.
Singh, J. , Kumar, D. , and Baleanu, D. , 2017, “ On the Analysis of Chemical Kinetics System Pertaining to a Fractional Derivative With Mittag-Leffler Type Kernel,” Chaos Interdiscip. J. Nonlinear Sci., 27(10), p. 103113. [CrossRef]
Wang, Z. , Huang, X. , and Shi, G. , 2011, “ Analysis of Nonlinear Dynamics and Chaos in a Fractional Order Financial System With Time Delay,” Comput. Math. Appl., 62(3), pp. 1531–1539. [CrossRef]
Li, C. , and Chen, G. , 2004, “ Chaos and Hyperchaos in the Fractional-Order Rössler Equations,” Physica A, 341, pp. 55–61. [CrossRef]
Grigorenko, I. , and Grigorenko, E. , 2003, “ Chaotic Dynamics of the Fractional Lorenz System,” Phys. Rev. Lett., 91(3), p. 034101. [CrossRef] [PubMed]
Hegazi, A. S. , and Matouk, A. E. , 2011, “ Dynamical Behaviors and Synchronization in the Fractional Order Hyperchaotic Chen System,” Appl. Math. Lett., 24(11), pp. 1938–1944. [CrossRef]
Wang, L. , and Xu, Y. , 2011, “ An Effective Hybrid Biogeography-Based Optimization Algorithm for Parameter Estimation of Chaotic Systems,” Expert Syst. Appl., 38(12), pp. 15103–15109. [CrossRef]
Cheng, C. H. , Cheng, T. Y. , Du, C. H. , Lu, Y. C. , Chiou, Y. P. , Liu, S. , and Wu, T. L. , 2014, “ Parameter Estimation of Chaotic Systems by an Oppositional Seeker Optimization Algorithm,” Nonlinear Dyn., 76(1), pp. 509–517. [CrossRef]
Li, X. , and Yin, M. , 2014, “ Parameter Estimation for Chaotic Systems by Hybrid Differential Evolution Algorithm and Artificial Bee Colony Algorithm,” Nonlinear Dyn., 77(1–2), pp. 61–71. [CrossRef]
Wang, J. , and Zhou, B. , 2016, “ A Hybrid Adaptive Cuckoo Search Optimization Algorithm for the Problem of Chaotic Systems Parameter Estimation,” Neural Comput. Appl., 27(6), pp. 1511–1517. [CrossRef]
Diethelm, K. , 1997, “ An Algorithm for the Numerical Solution of Differential Equations of Fractional Order,” Electron. Trans. Numer. Anal., 5(1), pp. 1–6. http://emis.ams.org/journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.pdf
Diethelm, K. , Ford, N. J. , and Freed, A. D. , 2002, “ A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations,” Nonlinear Dyn., 29(1), pp. 3–22. [CrossRef]
Baleanu, D. , Mousalou, A. , and Rezapour, S. , 2017, “ On the Existence of Solutions for Some Infinite Coefficient-Symmetric Caputo-Fabrizio Fractional Integro-Differential Equations,” Boundary Value Probl., 2017(1), p. 145. [CrossRef]
Kumar, D. , Singh, J. , and Baleanu, D. , 2018, “ A New Numerical Algorithm for Fractional Fitzhugh–Nagumo Equation Arising in Transmission of Nerve Impulses,” Nonlinear Dyn., 91(1), pp. 307–317. [CrossRef]
Parlitz, U. , Junge, L. , and Kocarev, L. , 1996, “ Synchronization-Based Parameter Estimation From Time Series,” Phys. Rev. E, 54(6), p. 6253. [CrossRef]
Parlitz, U. , 1996, “ Estimating Model Parameters From Time Series by Autosynchronization,” Phys. Rev. Lett., 76(8), p. 1232. [CrossRef] [PubMed]
Gao, F. , Fei, F.-X. , Lee, X.-J. , Tong, H.-Q. , Deng, Y.-F. , and Zhao, H.-L. , 2014, “ Inversion Mechanism With Functional Extrema Model for Identification Incommensurate and Hyper Fractional Chaos Via Differential Evolution,” Expert Syst. Appl., 41(4), pp. 1915–1927. [CrossRef]
Storn, R. , and Price, K. , 1997, “ Differential Evolution—A Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces,” J. Global Optim., 11(4), pp. 341–359. [CrossRef]
Shaefer, C. G. , 1993, “ Genetic Algorithm,” Rowland Institute for Science, Cambridge, MA, U.S. Patent No. US5255345A. https://patents.google.com/patent/US5255345A/en
Kennedy, J. , and Eberhart, R. , 1995, “ Particle Swarm Optimization (PSO),” IEEE International Conference on Neural Networks, Perth, Australia, Nov. 27–Dec. 1, pp. 1942–1948.
Yang, X. S. , and Deb, S. , 2009, “ Cuckoo Search Via Lévy Flights,” World Congress on Nature and Biologically Inspired Computing (NaBIC 2009), Coimbatore, India, Dec. 9–11, pp. 210–214.
Tang, Y. , Zhang, X. , Hua, C. , Li, L. , and Yang, Y. , 2012, “ Parameter Identification of Commensurate Fractional-Order Chaotic System Via Differential Evolution,” Phys. Lett. A, 376(4), pp. 457–464. [CrossRef]
Sun, J. , Xu, W. , and Feng, B. , 2005, “ A Global Search Strategy of Quantum-Behaved Particle Swarm Optimization,” IEEE Conference on Cybernetics and Intelligent Systems, Singapore, Dec. 1–3, pp. 111–116.
Zhang, X. F. , and Sui, G. F. , 2013, “ Quantum-Behaved Particle Swarm Optimization Algorithm for Solving Nonlinear Equations,” Adv. Mater. Res., 756–759, pp. 2926–2931.
Turgut, O. E. , Turgut, M. S. , and Coban, M. T. , 2014, “ Chaotic Quantum Behaved Particle Swarm Optimization Algorithm for Solving Nonlinear System of Equations,” Comput. Math. Appl., 68(4), pp. 508–530. [CrossRef]
Tang, D. , Cai, Y. , Zhao, J. , and Xue, Y. , 2014, “ A Quantum-Behaved Particle Swarm Optimization With Memetic Algorithm and Memory for Continuous Non-Linear Large Scale Problems,” Inf. Sci., 289(24), pp. 162–189. [CrossRef]
Samko, S. G. , Kilbas, A. A. , Marichev, O. I. , et al. ., 1993, “ Fractional Integrals and Derivatives,” Theory and Applications, Gordon and Breach, Yverdon, Switzerland.
Luchko, Y. , and Gorenflo, R. , 1999, “ An Operational Method for Solving Fractional Differential Equations With the Caputo Derivatives,” Acta Math. Vietnam., 24(2), pp. 207–233. http://journals.math.ac.vn/acta/pdf/9902207.pdf
Hairer, E. , Rsett, S. P. , and Wanner, G. , 1993, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed., Springer-Verlag, New York.
Hairer, E. , and Wanner, G. , 1991, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin.
Tizhoosh, H. R. , 2005, “ Opposition-Based Learning: A New Scheme for Machine Intelligence,” International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce, Vienna, Austria, Nov. 28–30, pp. 695–701.
Wang, H. , Wu, Z. , Rahnamayan, S. , Liu, Y. , and Ventresca, M. , 2011, “ Enhancing Particle Swarm Optimization Using Generalized Opposition-Based Learning,” Inf. Sci., 181(20), pp. 4699–4714. [CrossRef]
Rao, R. V. , Savsani, V. J. , and Vakharia, D. P. , 2011, “ Teaching-Learning-Based Optimization: A Novel Method for Constrained Mechanical Design Optimization Problems,” Comput.-Aided Des., 43(3), pp. 303–315. [CrossRef]
Hu, W. , Yu, Y. , and Zhang, S. , 2015, “ A Hybrid Artificial Bee Colony Algorithm for Parameter Identification of Uncertain Fractional-Order Chaotic Systems,” Nonlinear Dyn., 82(3), pp. 1441–1456. [CrossRef]
Zhu, W. , Fang, J. , Tang, Y. , Zhang, W. , and Xu, Y. , 2012, “ Identification of Fractional-Order Systems Via a Switching Differential Evolution Subject to Noise Perturbations,” Phys. Lett. A, 376(45), pp. 3113–3120. [CrossRef]
Huang, Y. , Guo, F. , Li, Y. , and Liu, Y. , 2015, “ Parameter Estimation of Fractional-Order Chaotic Systems by Using Quantum Parallel Particle Swarm Optimization Algorithm,” PloS One, 10(1), p. e0114910. [CrossRef] [PubMed]
Wei, J. , Yu, Y. , and Wang, S. , 2015, “ Parameter Estimation for Noisy Chaotic Systems Based on an Improved Particle Swarm Optimization Algorithm,” J. Appl. Anal. Comput., 5(2), pp. 232–242. http://jaac.ijournal.cn/ch/reader/create_pdf.aspx?file_no=20150207&journal_id=jaac
Hu, W. , Yu, Y. , and Wang, S. , 2015, “ Parameters Estimation of Uncertain Fractional-Order Chaotic Systems Via a Modified Artificial Bee Colony Algorithm,” Entropy, 17(12), pp. 692–709. [CrossRef]
Gu, W. , Yu, Y. , and Hu, W. , 2015, “ Parameter Estimation of Unknown Fractional-Order Memristor-Based Chaotic Systems by a Hybrid Artificial Bee Colony Algorithm Combined With Differential Evolution,” Nonlinear Dyn., 84(2), pp. 779–795. [CrossRef]
Chen, W. C. , 2008, “ Nonlinear Dynamics and Chaos in a Fractional-Order Financial System,” Chaos Solitons Fractals, 36(5), pp. 1305–1314. [CrossRef]
Petráš, I., and Bednárová, D., 2009, “ Fractional-Order Chaotic Systems,” IEEE International Conference on Emerging Technologies and Factory Automation, pp. 1031–1038.


Grahic Jump Location
Fig. 1

The objective function of fractional-order financial system (20)

Grahic Jump Location
Fig. 3

Evolution process of objective function values for fractional-order financial system (20)

Grahic Jump Location
Fig. 4

The objective function of fractional-order Rössler system (23)

Grahic Jump Location
Fig. 5

Evolution process of the estimated parameters for fractional-order Rössler system (23): (a) evolution process of a, (b) evolution process of b, (c) evolution process of c, (d) evolution process of q1, (e) evolution process of q2, and (f) evolution process of q3

Grahic Jump Location
Fig. 6

Evolution process of objective function values for fractional-order Rössler system (23)

Grahic Jump Location
Fig. 2

Evolution process of the estimated parameters for fractional-order financial system (20): (a) evolution process of a, (b) evolution process of b, (c) evolution process of c, (d) evolution process of q1, (e) evolution process of q2, and (f) evolution process of q3



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In