Nonlinear Dynamics

, Volume 89, Issue 3, pp 1889–1903 | Cite as

Impulsive stabilization of chaos in fractional-order systems

  • Marius-F. DancaEmail author
  • Michal Fečkan
  • Guanrong Chen
Original Paper


This paper considers a class of nonlinear impulsive Caputo differential equations of fractional order, which models chaotic systems. Computer-assisted proof of chaos suppression by stabilizing the unstable system equilibria is provided. A nonexistence result of periodic solutions is presented, and the commensurate fractional-order Lorenz system is simulated for illustration.


Impulsive Caputo differential equation of fractional order Adams–Bashforth–Moulton method Chaos stabilization 



The authors thank to Professor Julien Clinton Sprott for interesting discussions related to the energy approach. M.-F. Danca is supported by Tehnic B SRL. M. Fečkan is supported in part by the Slovak Research and Development Agency under the Contract No. APVV-14-0378 and by the Slovak Grant Agency VEGA Nos. 2/0153/16 and 1/0078/17. G. Chen is supported by the Hong Kong Research Grants Council under the GRF Grant CityU 11234916.


  1. 1.
    Yang, T.: Impulsive Control Theory. Springer, Berlin (2001)zbMATHGoogle Scholar
  2. 2.
    Oldham, K., Spainer, J.: Fractional Calculus. Academic press, Dordrecht (1974)Google Scholar
  3. 3.
    Podlubny, I.: Fractional Differential Equations. Academic Press, Dordrecht (1989)zbMATHGoogle Scholar
  4. 4.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. Series on Complexity Nonlinearity and Chaos, vol. 3. World Scientific, Singapore (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Sabatier, J., Ionescu, C., Tar, J.K., Teneiro, M.J.A.: guest editors. New challenges in fractional systems. (2013).
  6. 6.
    Riccardo, C., Trujillo, J.J., Machado, J.A.T.: Theory and applications of fractional order systems. Math. Probl. Eng. (2016). doi: 10.1155/2016/7903424 Google Scholar
  7. 7.
    Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Bifurcation and chaos in noninteger order cellular neural networks. Int. J. Bifurc. Chaos 8, 1527–1539 (1998). doi: 10.1142/S0218127498001170 CrossRefzbMATHGoogle Scholar
  8. 8.
    Boroomand, A., Menhaj, M.: Fractional-order Hopfield neural networks. In: Lecture Notes in Computer Science, vol. 5506, pp. 883–890 (2009)Google Scholar
  9. 9.
    Kaslik, E., Sivasundaram, 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. Real World Appl. 13, 1489–1497 (2012). doi: 10.1016/j.nonrwa.2011.11.013 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractal system as represented by singularity function. IEEE Xplore: IEEE Trans. Autom. Control 37, 1465–1470 (1992). doi: 10.1109/9.159595 MathSciNetzbMATHGoogle Scholar
  11. 11.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002). doi: 10.1023/A:1016592219341 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Scherer, R., Kalla, S.L., Tang, Y., Huang, J.: The Grünwald–Letnikov method for fractional differential equations. Comput. Math. Appl. 62, 902–917 (2011). doi: 10.1016/j.camwa.2011.03.054 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, G., Yu, X.: Chaos Control: Theory and Applications. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Oustaloup, A., Sabatier, J., Lanusse, P.: From fractal robustness to the CRONE control. Fract. Calc. Appl. Anal. 2, 1–30 (1999)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang, X.-Y., He, Y.-J., Wang, M.-J.: Chaos control of a fractional order modified coupled dynamos system. Nonlinear Anal. Theory Methods Appl. 71, 6126–6134 (2009). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ahmad, W.M., Harba, A.M.: On nonlinear control design for autonomous chaotic systems of integer and fractional orders. Chaos Soliton Fract. 18, 693–701 (2003). doi: 10.1016/S0960-0779(02)00644-6 CrossRefzbMATHGoogle Scholar
  17. 17.
    Boulkroune, A., Chekireb, H., Tadjine, M., Bouatmane, S.: Observer-based adaptive feedback controller of a class of chaotic systems. Int. J. Bifurc. Chaos 16, 3411–3419 (2006)Google Scholar
  18. 18.
    Yin, C., Chen, Y.Q., Zhong, S.: Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica 50, 3173–3181 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yin, C., Cheng, Y., Chen, Y.Q., Stark, B., Zhong, S.: Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems. Nonlinear Dyn. 82, 39–52 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, G., Dong, X.: From Chaos to Order-Methodologies Perspectives and Applications. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Richter, H., Reinschke, K.J.: Local control of chaotic systems—a Lyapunov approach. Int. J. Bifurc. Chaos 8, 1565–1573 (1998). doi: 10.1142/S0218127498001212 CrossRefzbMATHGoogle Scholar
  23. 23.
    Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992). doi: 10.1016/0375-9601(92)90745-8 CrossRefGoogle Scholar
  24. 24.
    Agarwal, R.P., Benchohra, M., Slimani, B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Benchohra, M., Slimani, B.A.: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equ. 2009, 1–11 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Benchohra, M., Seba, D.: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2009, 1–14 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Guan, Z.-H., Chen, G., Ueta, T.: On impulsive control of a periodically forced chaotic pendulum system. IEEE Trans. Autom. Control 45, 1724–1727 (2000). doi: 10.1109/9.880633 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Yang, Xujun, Li, Chuandong, Huang, Tingwen, Song, Qiankun: Mittag-Leffler stability analysis of nonlinear fractional-order systems with impulses. Appl. Math. Comput. 293, 416422 (2017)MathSciNetGoogle Scholar
  29. 29.
    Yang, Xujun, Li, Chuandong, Song, Qiankun, Huang, Tingwen, Chen, Xiaofeng: Mittag-Leffler stability analysis on variable-time impulsive fractional-order neural networks. Neurocomputing 207, 276286 (2016)Google Scholar
  30. 30.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. Series on Nonlinear Science. World Scientific, Singapore (1995)Google Scholar
  31. 31.
    Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Acta Rheologica 45, 765–771 (2006). doi: 10.1007/s00397-005-0043-5 CrossRefGoogle Scholar
  32. 32.
    Matignon, D.: Stability results of fractional differential equations with applications to control processing. In: IEEE-SMC Proceedings of the Computational Engineering in Systems and Application Multiconference. IMACS, vol. 2, pp. 963–968 (1996)Google Scholar
  33. 33.
    Tavazoei, M.S., Haeri, M.: Chaotic attractors in incommensurate fractional order systems. Phys. D 237, 2628–2637 (2008). doi: 10.1016/j.physd.2008.03.037 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Li, F., Yao, C.: The infinite-scroll attractor and energy transition in chaotic circuit. Nonlinear Dyn. 84, 2305–2315 (2016). doi: 10.1007/s11071-016-2646-z MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sarasola, C., Torrealdea, F.J., d’ Anjou, A., Moujahid, A., Graña, M.: Energy balance in feedback synchronization of chaotic systems. Phys. Rev. E 69, 011606 (2004). doi: 10.1103/PhysRevE.69.011606 CrossRefGoogle Scholar
  36. 36.
    Danca, M.-F., Kuznetsov, N., Chen, G.: Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system. Nonlinear Dyn. (2017). doi: 10.1007/s11071-016-3276-1
  37. 37.
    Wu, X.J., Shen, S.L.: Chaos in the fractional-order Lorenz system. Int. J. Comput. Math. 86, 1274–1282 (2009). doi: 10.1080/00207160701864426 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Li, D., Lu, J., Wu, X., Chen, G.: Estimating the bounds for the Lorenz family of chaotic systems. Chaos Solitons Fract. 23, 529–534 (2005). doi: 10.1016/j.chaos.2004.05.021 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Tavazoei, M., Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45, 1886–1890 (2009). doi: 10.1016/j.automatica.2009.04.001 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yazdani, M., Salarieh, H.: On the existence of periodic solutions in time-invariant fractional order systems. Automatica 47, 1834–1837 (2011). doi: 10.1016/j.automatica.2011.04.013 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kang, Y.-M., Xie, Y., Lu, J.C., Jiang, J.: On the nonexistence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical systems. Nonlinear Dyn. 82, 1259–1267 (2015). doi: 10.1007/s11071-015-2232-9 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Shen, J., Lam, J.: Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica 50, 547–551 (2014). doi: 10.1016/j.automatica.2013.11.018 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wang, J.R., Fečkan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, 806–831 (2016). doi: 10.1515/fca-2016-0044 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAvram Iancu University of Cluj-NapocaCluj-NapocaRomania
  2. 2.Romanian Institute of Science and TechnologyCluj-NapocaRomania
  3. 3.Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and InformaticsComenius University in BratislavaBratislavaSlovak Republic
  4. 4.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovak Republic
  5. 5.Department of Electronic EngineeringCity University of Hong KongHong KongChina

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