Skip to main content
SpringerLink
Log in

Stability analysis on a class of nonlinear fractional-order systems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The stability of a class of commensurate and incommensurate nonlinear fractional-order systems is studied. First, two comparison inequalities for incommensurate fractional-order systems are proposed. Based on that, a stability criterion regarding a class of incommensurate fractional-order system is given. And then, three stability criteria are presented concerning a typical class of commensurate nonlinear fractional-order systems. After that, two global stability criteria concerning commensurate and incommensurate nonlinear fractional-order systems are provided, respectively. Finally, the fractional-order Liu system and the fractional-order Chen system are taken as examples to show how to apply the proposed results to stabilization problems.

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
Fig. 5

Similar content being viewed by others

References

  1. Ndoye, I., Zasadzinski, M., Darouach, M., Radhy, N.-E.: Observer-based control for fractional-order continuous-time systems. In: Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, WeBIn5.3, pp. 1932–1937 (2009)

  2. Wang, Z.L., Yang, D.S., Ma, T.D., Sun, N.: Stability analysis for nonlinear fractional-order systems based on comparison principle. Nonlinear Dyn. 75, 387–402 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kaczorek, T.: Practical stability and asymptotic stability of positive fractional 2D linear systems. Asian J. Control 12(2), 200–207 (2010)

    Article  MathSciNet  Google Scholar 

  4. Ntouyas, S.K., Wang, G.T., Zhang, L.H.: Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments. Opusc. Math. 31(3), 433–442 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kaczorek, T.: Positive fractional linear electrical circuits. In: Proceedings of the Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2013, vol. 8903, p. 89031N-14 (2013)

  6. Kaczorek, T.: Stability tests of positive fractional continuous time linear systems with delays. Int. J. Mar. Navig. Saf. Sea Transp. 7(2), 211–215 (2013)

    Article  MATH  Google Scholar 

  7. Li, Y., Chen, Y.Q., Podlunby, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, Y., Chen, Y.Q., Podlunby, I.: Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput. Math. Appl. 59(5), 1810–1821 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yu, J.M., Hu, H., Zhou, S.B., Lin, X.R.: Generalized Mittag–Leffler stability of multi-variables fractional order nonlinear systems. Automatica 49(6), 1798–1803 (2013)

    Article  MathSciNet  Google Scholar 

  10. Baleanu, D., Sadati, S.J., Ghaderi, R., Ranjbar, A., Abdeljawad, T., Jarad, F.: Razumikhin stability theorem for fractional systems with delay. Abstr. Appl. Anal. 2010, 124812 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Rivero, M., Rogosin, S.V., Machado, J.A.T., Trujillo, J.J.: Stability of fractional order systems. Math. Prob. Eng. 2013, 356215 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Petráš, I.: Stability of fractional-order systems with rational orders: a survey. Fract. Calc. Appl. Anal. 12(3), 269–298 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67(4), 2433–2439 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, Y., Chen, Y. Q., Zhai, L.: Stability of fractional-order population growth model based on distributed-order approach. Proceedings of the 33rd Chinese Control Conference, pp. 2586–2591 (2014)

  15. Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order system. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)

    Article  MathSciNet  Google Scholar 

  16. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order system. Commun. Nonlinear Sci. Numer. Simul. 22, 650–659 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. J. Math. Anal. Appl. 325, 542–553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tavazoei, M.S., Haeri, M.: Chaotic attractors in incommensurate fractional order systems. Phys. D 237, 2628–2637 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tavazoei, M.S., Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45(8), 1886–1890 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kaslika, E., Sivasundaramc, S.: Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal. 13(3), 1489–1497 (2012)

    Article  MathSciNet  Google Scholar 

  21. Shen, J., Lam, J.: Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica 50(2), 547–551 (2014)

    Article  MathSciNet  Google Scholar 

  22. Deng, W.: Smoothness and stability of the solutions for nonlinear fractional differential equation. Nonlinear Anal. 72(3–4), 1768–1777 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, C.P., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71(4), 621–633 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wen, X.J., Wu, Z.M., Lu, J.G.: Stability analysis of a class of nonlinear fractional-order systems. IEEE Trans. Circ. Syst. II 55(11), 1178–1182 (2008)

    Article  Google Scholar 

  25. Chen, L.P., Chai, Y., Wu, R.C., Yang, J.: Stability and stabilization of a class of nonlinear fractional-order systems with Caputo derivative. IEEE Trans. Circ. Syst. II 59(9), 602–606 (2012)

  26. Chen, L.P., He, Y., Chai, Y., Wu, R.C.: New results on stability and stabilization of a class of nonlinear fractional-order systems. Nonlinear Dyn. 75(4), 633–641 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  28. Matignon, D.: Stability results on fractional differential equations with applications to control processing. In: Processdings of IMACS-SMC, Lille, France pp. 963–968 (1996)

  29. Deng, W., Li, C.P., Lü, J.: Stability analysis of linear fractional differential system with multiple delays. Nonlinear Dyn. 48(4), 409–416 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nersesov, S.G., Haddad, W.M.: On the stability and control of nonlinear dynamical systems via vector Lyapunov functions. IEEE Trans. Autom. Control 51, 203–215 (2006)

    Article  MathSciNet  Google Scholar 

  31. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  32. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  33. Lax, P.D.: Linear Algebra and Its Applications, 2nd edn. Wiley, New York (2007)

    MATH  Google Scholar 

  34. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91(3), 034101(1–4) (2003)

    Article  Google Scholar 

  35. Lu, J.G.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A 354, 305–311 (2006)

    Article  Google Scholar 

  36. Wang, X.Y., Wang, M.J.: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos 17(3), 033106(1–6) (2007)

    MATH  Google Scholar 

  37. Aguila-Camacho, N., Duarte-Mermoud, M.A.: Comments on “Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks”. Comm. Nonlinear Sci. Num. Simul. 25, 145–148 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Northeastern University, Shenyang, 110004, People’s Republic of China

    Zhiliang Wang, Dongsheng Yang & Huaguang Zhang

Authors
  1. Zhiliang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dongsheng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huaguang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiliang Wang.

Additional information

This work is supported by the Northeastern University Fundamental Research Grants (Nos. N140404017, N110404023), the National Natural Science Foundations of China (Nos. 60804006, 61273029, 61273027, 61203026, and 61104080), the Natural Science Foundation of Liaoning (2013020037) and the Program for New Century Excellent Talents in University, China (NCET-12-0106).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Yang, D. & Zhang, H. Stability analysis on a class of nonlinear fractional-order systems. Nonlinear Dyn 86, 1023–1033 (2016). https://doi.org/10.1007/s11071-016-2943-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2943-6

Keywords

Search

Navigation