Skip to main content
SpringerLink
Log in

Strange attractors generated by a fractional order switching system and its topological horseshoe

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

Abstract

Chaos generation in a new fractional order unstable dissipative system with only two equilibrium points is reported. Based on the integer version of an unstable dissipative system (UDS) and using the same system’s parameters, chaos behavior is observed with an order less than three, i.e., 2.85. The fractional order can be decreased as low as 2.4 varying the eigenvalues of the fractional UDS in accordance with a switching law that fulfills the asymptotic stability theorem for fractional systems. The largest Lyapunov exponent is computed from the numerical time series in order to prove the chaotic regime. Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order switching system avoiding the possible numerical bias of Lyapunov exponents. Finally, an electronic circuit is designed to synthesize this fractional order chaotic system.

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Grzesikiewicz, W., Wakulicz, A., Zbiciak, A.: Non-linear problems of fractional calculus in modeling of mechanical systems. Int. J. Mech. Sci. 70, 90–98 (2013)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  3. Diethelm, K.: The Analysis of Fractional Differential Equations, An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)

    MATH  Google Scholar 

  4. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls, Fundamentals and Applications. Springer, London (2010)

    Book  MATH  Google Scholar 

  5. Petras, I.: Fractional-Order Nonlinear Systems, Modeling, Analysis and Simulation. Higher Education Press and Springer, Beijing and Berlin (2011)

    Book  MATH  Google Scholar 

  6. Ghasemi, S., Tabesh, A., Askari-Marnani, J.: Application of fractional calculus theory to robust controller design for wind turbine generators. IEEE Trans. Energy Convers. 29, 780–787 (2014)

    Article  Google Scholar 

  7. Bhalekar, S.: Synchronization of incommensurate non-identical fractional order chaotic systems using active control. Eur. Phys. J. Spec. Top. 223, 1495–1508 (2014)

    Article  Google Scholar 

  8. Sasso, A., Palmieri, G., Amodio, D.: Application of fractional derivative models in linear viscoelastic problems. Mech. Time Depend. Mater. 15, 367–387 (2011)

    Article  Google Scholar 

  9. Guyomar, D., Ducharne, B., Sebald, G., Audiger, D.: Fractional derivative operators for modeling the dynamic polarization behavior as a function of frequency and electric field amplitude. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56, 437–443 (2009)

    Article  Google Scholar 

  10. Zhang, R., Yang, S.: Adaptive synchronization of fractional-order chaotic systems via a single driving variable. Nonlinear Dyn. 66, 831–837 (2011)

    Article  MATH  Google Scholar 

  11. Rakkiyappan, R., Velmurugan, G., Cao, J.: Stability analysis of memristor-based fractional-order neuronal networks with different memductance function. Cogn. Neurodyn. 9, 145–177 (2015)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Lu, J.G., Chen, G.: A note on the fractional order Chen system. Chaos Soliton Fractals 27, 685–688 (2006)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  17. Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Topological horseshoe analysis and circuit realization for a fractional-order Lü system. Nonlinear Dyn. 74, 203–212 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. HosseinNia, S.H., Magin, R.L., Vinagre, B.M.: Chaos in fractional and integer order NSG systems. Signal Process. 107, 302–311 (2015)

    Article  Google Scholar 

  19. Ma, T., Zhang, J.: Hybrid synchronization of coupled fractional-order complex networks. Neurocomputing 157, 166–172 (2015)

    Article  Google Scholar 

  20. Kiani-B, A., Fallahi, K., Pariz, N., Leung, H.: A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter. Commun. Nonlinear Sci. 14, 863–879 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Fast projective synchronization of fractional order chaotic and reverse chaotic systems with its application to an affine cipher using date of birth (DOB). Nonlinear Dyn. 80, 1883–1897 (2015)

    Article  MathSciNet  Google Scholar 

  22. Xu, Y., Wang, H., Li, Y., Pei, B.: Image encryption based on synchronization of fractional chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19, 3735–3744 (2014)

    Article  MathSciNet  Google Scholar 

  23. Muthukumar, P., Balasubramaniam, P.: Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography. Nonlinear Dyn. 74, 1169–1181 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES). Nonlinear Dyn. 77, 1547–1559 (2014)

    Article  MathSciNet  Google Scholar 

  25. Campos-Canton, E., Barajas-Ramirez, J.G., Solis-Perales, G., Femat, R.: Multiscroll attractors by switching systems. Chaos 20, 013116/6 (2010)

    Article  Google Scholar 

  26. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)

    Book  MATH  Google Scholar 

  27. Yang, X.S., Yu, Y.G., Zhang, S.C.: A new proof for existence of horseshoe in the Rössler system. Chaos Soliton Fractals 18, 223–227 (2003)

  28. Yang, X.S., Tang, Y.: Horseshoes in piecewise continuous maps. Chaos Soliton Fractals 19, 841–845 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Chaotic Characteristics analysis and circuit implementation for a fractional-order system. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 61, 845–853 (2014)

    Article  Google Scholar 

  30. Yang, X.S.: Topological horseshoes and computer assisted verification of chaotic dynamics. Int. J. Bifurc. Chaos 19, 1127–1145 (2009)

    Article  MATH  Google Scholar 

  31. Wu, W.J., Chen, Z.Q., Yuan, Z.Z.: A computer-assisted proof for the existence of horseshoe in a novel chaotic system. Chaos Soliton Fractals 41, 2756–2761 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Diethelm, K., Neville, F., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MATH  Google Scholar 

  33. Munoz-Pacheco, J.M., Zambrano-Serrano, E., Felix-Beltran, O.G., Gomez-Pavon, L.C., Luis-Ramos, A.: Synchronization of PWL function-based 2D and 3D multi-scroll chaotic systems. Nonlinear Dyn. 70, 1633–1643 (2012)

    Article  MathSciNet  Google Scholar 

  34. Munoz-Pacheco, J.M., Tlelo-Cuautle, E.: Simulation of Chua’s circuit by automatic control of step-size. Appl. Math. Comput. 190, 1526–1533 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mohammad, S.T., Mohammad, H.: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. 69, 1299–1320 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series methods: the TISEAN package. Chaos 9, 413 (1999)

    Article  MATH  Google Scholar 

  37. Krishna, B.T., Reddy, K.V.V.S.: Active and passive realization of fractance device of order 1/2. Active and Passive Electronic Components 2008, Article ID 369421 (2008)

Download references

Acknowledgments

E. Zambrano-Serrano is a doctoral fellow of CONACYT (Mexico) in the Graduate Program on Control and Dynamical Systems at DMAp-IPICYT. E. Campos-Cantón acknowledges CONACYT for the financial support through Project No. 181002. J.M. Muñoz-Pacheco thanks VIEP-BUAP (No. MUPJ-ING15-G) to partially support this research.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Instituto Potosino de Investigación Científica y Tecnológica, 78216, San Luis Potosí, SLP., Mexico

    E. Zambrano-Serrano & E. Campos-Cantón

  2. Faculty of Electronics Sciences, Benemérita Universidad Autónoma de Puebla, 72570, Puebla, PUE., Mexico

    J. M. Muñoz-Pacheco

Authors
  1. E. Zambrano-Serrano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Campos-Cantón
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. M. Muñoz-Pacheco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Zambrano-Serrano.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zambrano-Serrano, E., Campos-Cantón, E. & Muñoz-Pacheco, J.M. Strange attractors generated by a fractional order switching system and its topological horseshoe. Nonlinear Dyn 83, 1629–1641 (2016). https://doi.org/10.1007/s11071-015-2436-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2436-z

Keywords

Search

Navigation