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Nonlinear Dynamics

, Volume 57, Issue 1–2, pp 157–170 | Cite as

Chaos in the fractional-order Volta’s system: modeling and simulation

  • Ivo Petráš
Original Paper

Abstract

This paper deals with a new fractional-order chaotic system. It is based on the concept of Volta’s system, where the mathematical model of Volta’s system contains fractional-order derivatives. This system has simple structure and can display a double-scroll attractor. The behavior of the integer-order and the fractional-order Volta’s system with total order less than three which exhibits chaos is presented as well. Computer simulations are cross-verified by the numerical calculation and the Matlab/Simulink models.

Keywords

Chaos Attractor Fractional calculus Fractional-order Volta’s system 

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References

  1. 1.
    Akcay, H., Malti, R.: On the completeness problem for fractional rationals with incommensurable differentiation orders. In: Proceedings of the 17th World Congress IFAC, pp. 15367–15371, Seoul, Korea, July 6–11 (2008) Google Scholar
  2. 2.
    Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Bifurcation and chaos in noninteger order cellular neural networks. Int. J. Bifurc. Chaos 8(7), 1527–1539 (1998) zbMATHCrossRefGoogle Scholar
  3. 3.
    Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Nonlinear Noninteger Order Circuits and Systems—An Introduction. World Scientific, Singapore (2000) zbMATHGoogle Scholar
  4. 4.
    Ahmad, W.M.: Hyperchaos in fractional-order nonlinear systems. Chaos Solitons Fractals 26, 1459–1465 (2005) zbMATHCrossRefGoogle Scholar
  5. 5.
    Carlson, G.E., Halijak, C.A.: Approximation of fractional capacitors (1/s)1/n by a regular Newton process. In: Proceedings of the Sixth Midwest Symposium on Circuit Theory, Madison, Wisconsin, May 6–7 (1963) Google Scholar
  6. 6.
    Dorčák, L.: Numerical models for simulation of the fractional-order control systems. UEF-04-94, The Academy of Sciences, Inst. of Experimental Physic, Košice, Slovakia (1994) Google Scholar
  7. 7.
    Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, G.: Fractal systems as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circ. Syst. I: Fundam. Theory Appl. 49(3), 363–367 (2002) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional-order derivatives—An expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lu system. Physica A 353, 61–72 (2005) CrossRefGoogle Scholar
  11. 11.
    Deng, W.: Short memory principle and a predictor–corrector approach for fractional differential equations. J. Comput. Appl. Math. 206, 174–188 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Deng, W.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1522 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ford, N., Simpson, A.: The numerical solution of fractional differential equations: Speed versus accuracy. Numerical Analysis Report 385, Manchester Centre for Computational Mathematics, Manchester (2001) Google Scholar
  14. 14.
    Gao, X., Yu, J.: Chaos in the fractional-order periodically forced complex Duffing’s oscillators. Chaos Solitons Fractals 24, 1097–1104 (2005) zbMATHCrossRefGoogle Scholar
  15. 15.
    Hao, B.: Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific, Singapore (1989) zbMATHGoogle Scholar
  16. 16.
    Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos on a fractional Chua’s system. IEEE Trans. Circ. Syst. Theory Appl. 42(8), 485–490 (1995) CrossRefGoogle Scholar
  17. 17.
    Hwang, Ch., Leu, J.F., Tsay, S.Y.: A note on time-domain simulation of feedback fractional-order systems. IEEE Trans. Autom. Control 47(4), 625–631 (2002) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Li, Ch., Chen, G.: Chaos and hyperchaos in the fractional-order Rossler equations. Physica A 341, 55–61 (2004) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Li, Ch., Peng, G.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos Solitons Fractals 27, 685–688 (2006) zbMATHCrossRefGoogle Scholar
  21. 21.
    Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 26, 1125–1133 (2005) zbMATHCrossRefGoogle Scholar
  22. 22.
    Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Chua’s circuits with a piecewise-linear nonlinearity. Int. J. Mod. Phys. B 19(20), 3249–3259 (2005) zbMATHGoogle Scholar
  23. 23.
    Matignon, D.: Stability properties for generalized fractional differential systems. In: Proc. of Fractional Differential Systems: Models, Methods and Applications, vol. 5, pp. 145–158 (1998) Google Scholar
  24. 24.
    Nakagava, M., Sorimachi, K.: Basic characteristics of a fractance device. IEICE Trans. Fundam. E75-A(12), 1814–1818 (1992) Google Scholar
  25. 25.
    Nimmo, S., Evans, A.K.: The effects of continuously varying the fractional differential order of chaotic nonlinear systems. Chaos Solitons Fractals 10(7), 1111–1118 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) zbMATHGoogle Scholar
  27. 27.
    Oustaloup, A.: La Derivation Non Entiere: Theorie, Synthese et Applications. Hermes, Paris (1995) zbMATHGoogle Scholar
  28. 28.
    Petráš, I.: Control of fractional-order Chua’s system. J. Electr. Eng. 53(07–08), 219–222 (2002) Google Scholar
  29. 29.
    Petráš, I., Dorčák, L.: Fractional-order control systems: modelling and simulation. Fract. Calc. Appl. Anal. 6(2), 205–232 (2003) zbMATHMathSciNetGoogle Scholar
  30. 30.
    Petráš, I.: Method for simulation of the fractional-order chaotic systems. Acta Montanistica Slovaca 11(4), 273–277 (2006) Google Scholar
  31. 31.
    Petráš, I.: A note on the fractional-order Chua’s system. Chaos Solitons Fractals 38(1), 140–147 (2008) CrossRefGoogle Scholar
  32. 32.
    Petráš, I.: Digital fractional-order differentiator/integrator—IIR type. MathWorks, Inc.: http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=3672, visited: May 26 (2008)
  33. 33.
    Petráš, I.: Digital fractional-order differentiator/integrator—fir type. MathWorks, Inc.: http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=3673, visited: May 26 (2008)
  34. 34.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) zbMATHGoogle Scholar
  35. 35.
    Podlubny, I., Petráš, I., Vinagre, B.M., O’Leary, P., Dorčák, Ľ.: Analogue realization of fractional-order controllers. Nonlinear Dyn. 29(1–4), 281–296 (2002) zbMATHCrossRefGoogle Scholar
  36. 36.
    Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5(4), 367–386 (2002) zbMATHMathSciNetGoogle Scholar
  37. 37.
    Podlubny, I.: Fractional-order systems and PI λ D μ-controllers. IEEE Trans. Autom. Control 44(1), 208–213 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3(4), 359–386 (2000) zbMATHMathSciNetGoogle Scholar
  39. 39.
    Valerio, D.: Toolbox ninteger for Matlab, v.2.3 (September 2005). web: http://web.ist.utl.pt/duarte.valerio/ninteger/ninteger.htm, visited: May 23 (2008)
  40. 40.
    Vinagre, B.M., Chen, Y.Q., Petráš, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. J. Franklin Inst. 340, 349–362 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    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) CrossRefMathSciNetGoogle Scholar
  42. 42.
    Tavazoei, M.S., Haeri, M.: A necessary condition for double scroll attractor existence in fractional—Order systems. Phys. Lett. A 367, 102–113 (2007) CrossRefGoogle Scholar
  43. 43.
    Tavazoei, M.S., Haeri, M.: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. 69, 1299–1320 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Wang, J.C.: Realizations of generalized Warburg impedance with RC ladder networks and transmission lines. J. Electrochem. Soc. 134(8), 1915–1920 (1987) CrossRefGoogle Scholar
  45. 45.
    Westerlund, S.: Dead Matter Has Memory! Causal Consulting, Kalmar (2002) Google Scholar
  46. 46.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institute of Control and Informatization of Production Processes, BERG FacultyTechnical University of KošiceKošiceSlovak Republic

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