Skip to main content
SpringerLink
Log in

Stability analysis of fractional differential equations with the short-term memory property

  • Original Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

The commonly defined fractional derivatives, like Riemann-Liouville and Caputo ones, are non-local operators which have the long-term memory characteristic, since they are in connection with all historical data. Because of this special property, they may be invalid for modeling some processes and materials with short-term memory phenomena. Motivated by this observation and in order to enlarge the applicability of fractional calculus theories, a fractional derivative with the short-term memory property is defined in this paper. It can be viewed as an extension of the Caputo fractional derivative. Several properties of this short memory fractional derivative are given and proved. Meanwhile, the stability problem for fractional differential equations with such a derivative is studied. By applying fractional Lyapunov direct methods, the stability conditions applicable to the local case and the global case are established respectively. Finally, three numerical examples are provided to demonstrate the correctness and effectiveness of the theoretical results.

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

Access this article

Log in via an institution

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

References

  1. Abdelouahab, M.S., Hamri, N.E.: The Grünwald-Letnikov fractional-order derivative with fixed memory length. Mediterranean Journal of Mathematics 13(2), 557–572 (2016)

    Article  MathSciNet  Google Scholar 

  2. Abedini, M., Nojoumian, M.A., Salarieh, H., Meghdari, A.: Model reference adaptive control in fractional order systems using discrete-time approximation methods. Communications in Nonlinear Science and Numerical Simulation 25(1–3), 27–40 (2015)

    Article  MathSciNet  Google Scholar 

  3. Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation 19(9), 2951–2957 (2014)

    Article  MathSciNet  Google Scholar 

  4. Averbach, E., Coriell, A.S.: Short-term memory in vision. The Bell System Technical Journal 40(1), 309–328 (1961)

    Article  Google Scholar 

  5. Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent-II. Geophysical Journal of the Royal Astronomical Society 13(5), 529–539 (1967)

    Article  Google Scholar 

  6. Deng, W.H.: Short memory principle and a predictor-corrector approach for fractional differential equations. Journal of Computational and Applied Mathematics 206(1), 174–188 (2007)

    Article  MathSciNet  Google Scholar 

  7. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics 29(1–4), 3–22 (2002)

    Article  MathSciNet  Google Scholar 

  8. Feckan, M., Wang, J.R.: Periodic impulsive fractional differential equations. Advances in Nonlinear Analysis 8(1), 482–496 (2019)

    Article  MathSciNet  Google Scholar 

  9. Gabano, J.D., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Processing 91(3), 531–541 (2011)

    Article  Google Scholar 

  10. Khalil, H.K.: Nonlinear Control. Pearson Higher Ed. (2014)

  11. Khalil, H.K.: Nonlinear Systems. Prentice-Hall, Upper Saddle River, NJ (2002)

  12. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier (2006)

  13. Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. John Willey, New York (1995)

    MATH  Google Scholar 

  14. Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers and Mathematics with Applications 59(5), 1810–1821 (2010)

    Article  MathSciNet  Google Scholar 

  15. Liao, Z., Peng, C., Peng, Y.: Subspace identification in time-domain for fractional order systems based on short memory principle. Journal of Applied Sciences 29(2), 209–215 (2011)

    Google Scholar 

  16. Liu, L., Pan, F., Xue, D.Y.: Variable-order fuzzy fractional PID controller. ISA Transactions 55, 227–233 (2015)

    Article  Google Scholar 

  17. Lin, J., Poinot, T., Trigeassou, J.C., Kabbaj, H., Faucher, J.: Modélisation et identification d\({^{\prime }}\)ordre non entier d\({^{\prime }}\)une machine asynchrone. Conférence Internationale Francophone d\({^{\prime }}\)Automatique (2000)

  18. Ma, Z.E., Zhou, Y.C., Li, C.Z.: Qualitative and Stability Methods for Ordinary Differential Equations (In Chinese). Science Press (2015)

  19. Magin, R.L.: Fractional calculus models of complex dynamics in biological tissues. Computers and Mathematics with Applications 59(5), 1586–1593 (2010)

    Article  MathSciNet  Google Scholar 

  20. Matignon, D.: Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications 2, 963–968 (1996)

    Google Scholar 

  21. Morgado, M.L., Ford, N.J., Lima, P.M.: Analysis and numerical methods for fractional differential equations with delay. Journal of Computational and Applied Mathematics 252, 159–168 (2013)

    Article  MathSciNet  Google Scholar 

  22. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  23. Patnaik, S., Hollkamp, J.P., Semperlotti, F.: Applications of variable-order fractional operators: a review. Proc. of the Royal Society A - Mathematical Physical and Engineering Sciences 476(2234) (2020)

  24. Petras, I.: Fractional-Order Nonlinear Systems: Modeling. Analysis and Simulation. Springer-Verlag, Berlin (2011)

    Book  Google Scholar 

  25. Picozzi, S., West, B.J.: Fractional Langevin model of memory in financial markets. Physical Review E 66(4), (2002)

  26. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, New York (1999)

  27. Podlubny, I.: Numerical solution of ordinary fractional differential equations by the fractional difference method. Advances in Difference Equations, 507–515 (1997)

  28. Ross, B.: The development of fractional calculus 1695–1900. Historia Mathematica 4(1), 75–89 (1977)

    Article  MathSciNet  Google Scholar 

  29. Sabatier, J., Aoun, M., Oustaloup, A., Gregoire, G., Ragot, F., Roy, P.: Fractional system identification for lead acid battery state of charge estimation. Signal Processing 86(10), 2645–2657 (2006)

    Article  Google Scholar 

  30. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)

    MATH  Google Scholar 

  31. Slotine, J.J.E., Li, W.P.: Applied Nonlinear Control. Prentice-Hall (1991)

  32. Song, C., Cao, J.D.: Dynamics in fractional-order neural networks. Neurocomputing 142, 494–498 (2014)

    Article  Google Scholar 

  33. Teodoro, G.S., Machado, J.A.T., Oliveira, E.C.: A review of definitions of fractional derivatives and other operators. Journal of Computational Physics 388, 195–208 (2019)

    Article  MathSciNet  Google Scholar 

  34. Than, H.T., Siegmund, S.: Stability of scalar nonlinear fractional differential equations with linearly dominated delay. Fractional Calculus and Applied Analysis 23(1), 250–267 (2020). https://doi.org/10.1515/fca-2020-0010

    Article  MathSciNet  MATH  Google Scholar 

  35. Todd, J.J., Marois, R.: Capacity limit of visual short-term memory in human posterior parietal cortex. Nature 428(6984), 751–754 (2004)

    Article  Google Scholar 

  36. Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. Dover Publications, New York (1959)

    MATH  Google Scholar 

  37. Wang, J.L., Li, H.F.: Surpassing the fractional derivative: Concept of the memory-dependent derivative. Computers and Mathematics with Applications 62(3), 1562–1567 (2011)

    Article  MathSciNet  Google Scholar 

  38. Wang, J.L., Li, H.F.: Memory-dependent derivative versus fractional derivative (II): Remodelling diffusion process. Applied Mathematics and Computation 391 (2021)

  39. Wang, D.L., Xiao, A.G., Liu, H.L.: Dissipativity and stability analysis for fractional functional differential equations. Fractional Calculus and Applied Analysis 18(6), 1399–1422 (2015). https://doi.org/10.1515/fca-2015-0081

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, H., Yu, Y.G., Wen, G.G., Zhang, S., Yu, J.Z.: Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154(22), 15–23 (2015)

    Article  Google Scholar 

  41. Wei, Y.H., Chen, Y.Q., Cheng, S.S., Wang, Y.: A note on short memory principle of fractional calculus. Fractional Calculus and Applied Analysis 20(6), 1382–1404 (2017). https://doi.org/10.1515/fca-2017-0073

    Article  MathSciNet  MATH  Google Scholar 

  42. Wu, G.C., Deng, Z.G., Baleanu, D., Zeng, D.Q.: New variable-order fractional chaotic systems for fast image encryption. Chaos 29(8), (2019)

  43. Wu, G.C., Luo, M.K., Huang, L.L., Banerjee, S.: Short memory fractional differential equations for new memristor and neural network design. Nonlinear Dynamics 100(4), 3611–3623 (2020)

    Article  Google Scholar 

  44. Wu, G.C., Zeng, D.Q., Baleanu, D.: Fractional impulsive differential equations: Exact solutions, integral equations and short memory case. Fractional Calculus and Applied Analysis 22(1), 180–192 (2019). https://doi.org/10.1515/fca-2019-0012

    Article  MathSciNet  MATH  Google Scholar 

  45. Wu, Z.J., Xia, Y.Q., Xie, X.J.: Stochastic Barbalat’s Lemma and Its Applications. IEEE Transactions on Automatic Control 57(6), 1537–1543 (2012)

    Article  MathSciNet  Google Scholar 

  46. Xu, Y.F., He, Z.M.: The short memory principle for solving Abel differential equation of fractional order. Computers and Mathematics with Applications 62(12), 4796–4805 (2011)

    Article  MathSciNet  Google Scholar 

  47. Xue, D.Y.: Fractional-Order Control Systems: Fundamentals and Numerical Implementations. De Gruyter, Berlin (2017)

    Book  Google Scholar 

  48. Yin, C., Huang, X.G., Dadras, S., Cheng, Y.H., Cao, J.W., Malek, H., Mei, J.: Design of optimal lighting control strategy based on multi-variable fractional-order extremum seeking method. Information Sciences 465, 38–60 (2018)

    Article  MathSciNet  Google Scholar 

  49. Zhang, S., Yu, Y.G., Wang, Q.: Stability analysis of fractional-order Hopfield neural networks with discontinuous activation functions. Neurocomputing 171, 1075–1084 (2016)

    Article  Google Scholar 

  50. Zhao, J.H., Zheng, L.C., Zhang, X.X., Liu, F.W.: Unsteady natural convection boundary layer heat transfer of fractional Maxwell viscoelastic fluid over a vertical plate. International Journal of Heat and Mass Transfer 97, 760–766 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (No. 2020YJS189), the Natural Science Foundation of Beijing Municipality (No. Z180005) and the National Nature Science Foundation of China (No. 61772063).

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Xudong Hai, Yongguang Yu, Conghui Xu & Guojian Ren

Authors
  1. Xudong Hai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongguang Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Conghui Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guojian Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guojian Ren.

Ethics declarations

Conflict of interest

The authors declare that they 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.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hai, X., Yu, Y., Xu, C. et al. Stability analysis of fractional differential equations with the short-term memory property. Fract Calc Appl Anal 25, 962–994 (2022). https://doi.org/10.1007/s13540-022-00049-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13540-022-00049-9

Keywords

Mathematics Subject Classification

Search

Navigation