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Topological horseshoe analysis and circuit realization for a fractional-order Lü system

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Abstract

The paper first discusses a newly reported fractional-order Lü system of order as low as 2.7 and shows its chaotic characteristics by numerical simulations. Then by using the topological horseshoe theory and computer-assisted proof, the existence of chaos in the system is verified theoretically. Finally, an analog hardware circuit is made for the fractional-order system, and the observed results demonstrate that the fractional-order Lü system is chaotic in physical experiment.

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References

  1. 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(8), 485–490 (1995)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  4. Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971)

    Google Scholar 

  5. Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode processes. J. Electroanal. Chem. 33, 253–265 (1971)

    Article  Google Scholar 

  6. Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304–311 (1991)

    Article  Google Scholar 

  7. Sugimoto, N.: Burgers equation with a fractional derivative: hereditary effects on nonlinear acoustic wave. J. Fluid Mech. 25, 631–653 (1991)

    Article  MathSciNet  Google Scholar 

  8. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real material. J. Appl. Mech. 51(2), 294–298 (1984)

    Article  MATH  Google Scholar 

  9. Lu, J.G., Chen, G.R.: A note on the fractional-order Chen system. Chaos Solitons Fractals 27, 685–688 (2006)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  11. Ge, Z.M., Qu, C.Y.: Chaos in a fractional order modified Duffing system. Chaos Solitons Fractals 34, 262–291 (2007)

    Article  MATH  Google Scholar 

  12. Huang, X., Zhao, Z., Wang, Z., Li, Y.X.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94, 13–21 (2012)

    Article  Google Scholar 

  13. Ahmad, W.M.: Hyperchaos in fractional order nonlinear systems. Chaos Solitons Fractals 27, 1459–1465 (2005)

    Article  Google Scholar 

  14. Chen, L.P., Chai, Y., Wu, R.W., Sun, J., Ma, T.D.: Cluster synchronization in fractional-order complex dynamical networks. Phys. Lett. A 376, 2381–2388 (2012)

    Article  MATH  Google Scholar 

  15. Wang, Z., Huang, X., Zhao, Z.: Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives. Nonlinear Dyn. 69, 999–1007 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, H.Q., Liao, X.F., Lou, M.W.: A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation. Nonlinear Dyn. 68, 137–149 (2012)

    Article  MATH  Google Scholar 

  17. Wu, X.G., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009)

    Article  MATH  Google Scholar 

  18. Zhang, R.Z., Yang, S.P.: Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller. Nonlinear Dyn. 68, 45–51 (2012)

    Article  MATH  Google Scholar 

  19. Pan, L., Zhou, W.N., Zhou, L., Sun, K.H.: Chaos synchronization between two different fractional-order hyperchaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16, 2628–2640 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, X.S.: Metric horseshoes. Chaos Solitons Fractals 20, 1149–1156 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yang, X.S., Li, Q.D.: Existence of horseshoe in a foodweb model. Int. J. Bifurc. Chaos 14, 1847–1852 (2004)

    Article  MATH  Google Scholar 

  23. Huang, Y., Yang, X.S.: Horseshoes in modified Chen’s attractors. Chaos Solitons Fractals 26, 79–85 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, X.S., Li, Q.D.: A computer-assisted proof of chaos in Josephson junctions. Chaos Solitons Fractals 27, 25–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yuan, Q., Li, Q.D., Yang, X.S.: Horseshoe chaos in a class of simple Hopfield neural networks. Chaos Solitons Fractals 39, 1522–1529 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  29. Yang, X.S., Li, Q.D., Chen, G.R.: A twin-star hyperchaotic attractor and its circuit implementation. Int. J. Circuit Theory Appl. 31, 637–640 (2003)

    Article  Google Scholar 

  30. Li, Y.X., Tang, W.K.S., Chen, G.R.: Hyperchaos evolved from the generalized Lorenz equation. Int. J. Circuit Theory Appl. 33, 235–251 (2005)

    Article  MATH  Google Scholar 

  31. Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Topological horseshoe analysis and the circuit implementation for a four-wing chaotic attractor. Nonlinear Dyn. 65, 131–140 (2011)

    Article  MATH  Google Scholar 

  32. Yu, S.M., Lü, J.H., Chen, G.R.: A family of n-scroll hyperchaotic attractors and their realization. Phys. Lett. A 364, 244–251 (2007)

    Article  Google Scholar 

  33. Jia, H.Y., Chen, Z.Q., Yuan, Z.Z.: A novel one equilibrium hyper-chaotic system generated upon Lü attractor. Chin. Phys. B 19(2), 020507 (2010)

    Article  Google Scholar 

  34. Qi, G.Y., Chen, G.R.: Analysis and circuit implementation of a new 4D chaotic system. Phys. Lett. A 352, 386–397 (2006)

    Article  MATH  Google Scholar 

  35. Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lü system. Phys. Lett. A 353, 61–72 (2005)

    Google Scholar 

  36. Tavazoei, M.S., Haeri, M.: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. 69, 1299–1320 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tavazoei, M.S., Haeri, M.: Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Process. 1(4), 171–181 (2007)

    Article  MathSciNet  Google Scholar 

  38. Tavazoei, M.S., Haeri, M.: A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367, 102–113 (2007)

    Article  MATH  Google Scholar 

  39. Deng, W.H.: Short memory principle and a predictor–corrector approach for fractional differential equations. J. Comput. Appl. Math. 206, 174–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Deng, W.H.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Petras, I.: Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. 57(1–2), 157–170 (2009)

    Article  MATH  Google Scholar 

  42. Charef, A., Sun, Y.Y., Tsao, Y.Y., Onaral, B.: Fractal systems as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11202148), the National Natural Science Foundation of China (Grant No. 61174094), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090031110029).

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Authors and Affiliations

  1. Department of Automation, Tianjin University of Science and Technology, Tianjin, 300222, P.R. China

    Hong-Yan Jia

  2. Department of Automation, Nankai University, Tianjin, 300071, P.R. China

    Zeng-Qiang Chen

  3. F’SATI/Department of Electrical Engineering, Tshwane University of Technology, Pretoria, 0001, South Africa

    Guo-Yuan Qi

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  1. Hong-Yan Jia
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  2. Zeng-Qiang Chen
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  3. Guo-Yuan Qi
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Correspondence to Hong-Yan Jia.

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Jia, HY., Chen, ZQ. & Qi, GY. Topological horseshoe analysis and circuit realization for a fractional-order Lü system. Nonlinear Dyn 74, 203–212 (2013). https://doi.org/10.1007/s11071-013-0958-9

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