Skip to main content
SpringerLink
Log in

Dissipativity and contractivity for fractional-order systems

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

Abstract

This paper concerns the dissipativity and contractivity of the Caputo fractional initial value problems. We prove that the systems have an absorbing set under the same assumptions as the classic integer-order systems. This directly extends the dissipativity from integer-order systems to the Caputo fractional-order ones. The fractional dissipativity conditions can be satisfied by many fractional chaotic systems and the systems from the spatial discretization of some time-fractional partial differential equations. The fractional-order systems satisfying the so-called one-sided Lipschitz condition are also considered in a similar way, and the contractivity property of their solutions is proved. Two numerical examples are provided to illustrate 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
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, London (1998)

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  4. Ozalp, N., Demirci, E.: A fractional order SEIR model with vertical transmission. Math. Comput. Model. 54(1), 1–6 (2011)

    Article  MathSciNet  Google Scholar 

  5. Ding, Y., Ye, H.: A fractional-order differential equation model of HIV infection of CD4\(\langle \sup \rangle \,{+}\,\langle /\sup \rangle \) T-cells. Math. Comput. Model. 50(3), 386–392 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Matignon, D.: Stability results for fractional differential equations with applications to control processing. Comput. Eng. Syst. Appl. 2, 963–968 (1996)

    Google Scholar 

  7. Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27–47 (2011)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Ibrahim, R.W.: Stability and stabilizing of fractional complex Lorenz systems. Abstract and Applied Analysis, ID: 127103 (2013)

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Bandyopadhyay, B., Kamal S.: Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach. Lecture Notes in Electrical Engineering. Springer, Switzerland (2015)

  12. Kamal, S., Raman, A., Bandyopadhyay, B.: Finite-time stabilization of fractional order uncertain chain of integrator: an integral sliding mode approach. Autom. Control IEEE Trans. 58(6), 1597–1602 (2013)

  13. Hale, J.K.: Asymptotic Behaviour of Dissipative Systems. American Mathematical Society, New York (1998)

    Google Scholar 

  14. Hill, A.T.: Global dissipativity for A-stable methods. SIAM J. Numer. Anal. 34(1), 119–142 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Humphries, A.R., Stuart, A.M.: Runge–Kutta methods for dissipative and gradient dynamical systems. SIAM J. Numer. Anal. 31(5), 1452–1485 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hartley, T.T., Lorenzo, C.F., Killory, Q.H.: Chaos in a fractional order Chua’s system. Circuits Syst. I Fundam. Theory Appl. IEEE Trans. 42(8), 485–490 (1995)

    Article  Google Scholar 

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

  18. Li, C., Peng, G.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22(2), 443–450 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16(2), 339–351 (2003)

  20. Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Phys. A Stat. Mech. Its Appl. 341, 55–61 (2004)

    Article  Google Scholar 

  21. Danca, M.F., Garrappa, R., Tang, W.K.S., Chen, G.: Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching. Comput. Math. Appl. 66(5), 702–716 (2013)

    Article  MathSciNet  Google Scholar 

  22. Pu, X., Guo, B.: Well-posedness and dynamics for the fractional Ginzburg–Landau equation. Appl. Anal. 92(2), 318–334 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Equations. Springer Series in Computational Mathematics, vol. 14, 2nd edn. Springer, Berlin (1996)

    Google Scholar 

  24. Stuart, A., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  25. Huang, C.: Dissipativity of Runge–Kutta methods for dynamical systems with delays. IMA J. Numer. Anal. 20(1), 153–166 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wen, L., Yu, Y., Wang, W.: Generalized Halanay inequalities for dissipativity of Volterra functional differential equations. J. Math. Anal. Appl. 347(1), 169–178 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Dahlquist, G.: Error Analysis for a Class of Methods for Stiff Non-linear Initial Value Problems. Numerical Analysis. Springer, Berlin (1976)

    Google Scholar 

  28. Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2951–2957 (2014)

    Article  MathSciNet  Google Scholar 

  29. Wei, Z., Li, Q., Che, J.: Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative. J. Math. Anal. Appl. 367(1), 260–272 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Chen, F., Nieto, J.J., Zhou, Y.: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. Real World Appl. 13(1), 287–298 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tarasov, V.E.: Fractional generalization of gradient systems. Lett. Math. Phys. 73(1), 49–58 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Ma, J., Chen, Y.: Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I). Appl. Math. Mech. 22(11), 1240–1251 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  34. Chen, J., Liu, F., Anh, V.: Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338(2), 1364–1377 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Stamova, I., Stamov, G.: Lipschitz stability criteria for functional differential systems of fractional order. J. Math. Phys. 54(4), 043502 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank the referees for their very valuable comments and suggestions. This work is supported by projects NSF of China (Nos. 11271311, 11426178), the Research Foundation of Education Commission of Hunan Province of China (No. 14A146), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 14JK1734).

Author information

Authors and Affiliations

  1. Department of Mathematics and Center for Nonlinear Studies, Northwest University, Xi’an, 710127, Shaanxi, People’s Republic of China

    Dongling Wang

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Aiguo Xiao

Authors
  1. Dongling Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aiguo Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongling Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, D., Xiao, A. Dissipativity and contractivity for fractional-order systems. Nonlinear Dyn 80, 287–294 (2015). https://doi.org/10.1007/s11071-014-1868-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1868-1

Keywords

Search

Navigation