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Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle

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  • Li, Hang
  • Shen, Yongjun
  • Han, Yanjun
  • Dong, Jinlu
  • Li, Jian

Abstract

Lyapunov exponents provide quantitative evidence for determining the stability and classifying the limit set of dynamical systems. There are several well-established techniques to compute Lyapunov exponent of integer-order systems, however, these techniques failed to generalize to fractional-order systems due to the nonlocality of fractional-order derivatives. In this paper, a method for determining the Lyapunov exponent spectrum of fractional-order systems is proposed. The proposed method is rigorously derived based on the memory principle of Grünwald–Letnikov derivative so that it is generally applicable and even well compatible with integer-order systems. Three classical examples, which are the fractional-order Lorenz system, fractional-order Duffing oscillator, and 4-dimensional fractional-order Chen system, are respectively employed to demonstrate the effectiveness of the proposed method for incommensurate, nonautonomous and low effective order systems as well as hyperchaotic systems. The simulation results suggest that the proposed method is indeed superior to the existing methods in accuracy and correctness.

Suggested Citation

  • Li, Hang & Shen, Yongjun & Han, Yanjun & Dong, Jinlu & Li, Jian, 2023. "Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    • Handle: RePEc:eee:chsofr:v:168:y:2023:i:c:s0960077923000681
    DOI: 10.1016/j.chaos.2023.113167
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    1. Zhou, Shuang & Wang, Xingyuan, 2021. "Simple estimation method for the largest Lyapunov exponent of continuous fractional-order differential equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    2. Higazy, M., 2020. "Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    3. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    4. Naim, Mouhcine & Lahmidi, Fouad & Namir, Abdelwahed & Kouidere, Abdelfatah, 2021. "Dynamics of an fractional SEIR epidemic model with infectivity in latent period and general nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    5. Wang, Sha & Yu, Yongguang & Diao, Miao, 2010. "Hybrid projective synchronization of chaotic fractional order systems with different dimensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4981-4988.
    6. Andrzej Stefanski & Tomasz kapitaniak, 2000. "Using chaos synchronization to estimate the largest lyapunov exponent of nonsmooth systems," Discrete Dynamics in Nature and Society, Hindawi, vol. 4, pages 1-9, January.
    7. Kamal, F.M. & Elsonbaty, A. & Elsaid, A., 2021. "A novel fractional nonautonomous chaotic circuit model and its application to image encryption," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    8. Sun, Lin & Chen, Yiming, 2021. "Numerical analysis of variable fractional viscoelastic column based on two-dimensional Legendre wavelets algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    9. Leng, Xiangxin & Gu, Shuangquan & Peng, Qiqi & Du, Baoxiang, 2021. "Study on a four-dimensional fractional-order system with dissipative and conservative properties," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
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