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Mathematical Problems in Engineering
Volume 2013, Article ID 380436, 7 pages
http://dx.doi.org/10.1155/2013/380436
Research Article

Elegant Chaos in Fractional-Order System without Equilibria

Dipartimento Ingegneria Innovazione, Università del Salento, 73100 Lecce, Italy

Received 16 July 2013; Accepted 30 November 2013

Academic Editor: Rafael Martínez-Guerra

Copyright © 2013 Donato Cafagna and Giuseppe Grassi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Cafagna, “Fractional calculus: a mathematical tool from the past for present engineers,” IEEE Industrial Electronics Magazine, vol. 1, no. 2, pp. 35–40, 2007. View at Publisher · View at Google Scholar · View at Scopus
  2. J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. H. Sun, A. Abdelwahab, and B. Onaral, “Linear approximation for transfer function with a pole of fractional order,” IEEE Transactions on Automatic Control, vol. 29, no. 5, pp. 441–444, 1984. View at Google Scholar · View at Scopus
  4. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  6. P. Arena, R. Caponetto, L. Fortuna, and D. Porto, Nonlinear Noninteger Order Circuits and Systems—An Introduction, vol. 38, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Rivero, S. V. Rogosin, J. A. Tenreiro Machado, and J. J. Trujillo, “Stability of fractional order systems,” Mathematical Problems in Engineering, vol. 2013, Article ID 356215, 14 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. S. Abd-Elouahab, N.-E. Hamri, and J. Wang, “Chaos control of a fractional-order financial system,” Mathematical Problems in Engineering, vol. 2010, Article ID 270646, 18 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. C. M. A. Pinto and J. A. Tenreiro Machado, “Complex order van der Pol oscillator,” Nonlinear Dynamics, vol. 65, no. 3, pp. 247–254, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  10. L. O. Chua, M. Komuro, and T. Matsumoto, “The double scroll family,” IEEE Transactions on Circuits and Systems I, vol. 33, no. 11, pp. 1072–1118, 1986. View at Google Scholar · View at Scopus
  11. T. Hartley, C. Lorenzo, and H. Qammer, “Chaos in a fractional order Chua's system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485–490, 1995. View at Publisher · View at Google Scholar · View at Scopus
  12. C. P. Li, W. H. Deng, and D. Xu, “Chaos synchronization of the Chua system with a fractional order,” Physica A, vol. 360, no. 2, pp. 171–185, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. Cafagna and G. Grassi, “Fractional-order Chua's circuit: time-domain analysis, bifurcation, chaotic behavior and test for chaos,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 3, pp. 615–639, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 9, no. 7, pp. 1465–1466, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. G. Lu and G. Chen, “A note on the fractional-order Chen system,” Chaos, Solitons and Fractals, vol. 27, no. 3, pp. 685–688, 2006. View at Publisher · View at Google Scholar · View at Scopus
  16. D. Cafagna and G. Grassi, “Bifurcation and chaos in the fractional-order Chen system via a time-domain approach,” International Journal of Bifurcation and Chaos, vol. 18, no. 7, pp. 1845–1863, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. D. Cafagna and G. Grassi, “Hyperchaos in the fractional-order Rössler system with lowest-order,” International Journal of Bifurcation and Chaos, vol. 19, no. 1, pp. 339–347, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. W. Deng and J. Lü, “Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control,” Chaos, vol. 16, no. 4, Article ID 043120, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. O. E. Rössler, “An equation for hyperchaos,” Physics Letters A, vol. 71, no. 2-3, pp. 155–157, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. Cafagna and G. Grassi, “Hyperchaotic coupled Chua circuits: an approach for generating new nxm-scroll attractors,” International Journal of Bifurcation and Chaos, vol. 13, no. 9, pp. 2537–2550, 2003. View at Google Scholar
  21. S. Jafari, J. C. Sprott, and S. M. R. Hashemi Golpayegani, “Elementary quadratic chaotic flows with no equilibria,” Physics Letters A, vol. 377, no. 9, pp. 699–702, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Z. Wei, “Dynamical behaviors of a chaotic system with no equilibria,” Physics Letters A, vol. 376, no. 2, pp. 102–108, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Z. Wang, S. Cang, E. O. Ochola, and Y. Sun, “A hyperchaotic system without equilibrium,” Nonlinear Dynamics, vol. 69, no. 1-2, pp. 531–537, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer, New York, NY, USA, 1983. View at MathSciNet
  25. H. Li, X. Liao, and M. Luo, “A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation,” Nonlinear Dynamics, vol. 68, no. 1-2, pp. 137–149, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific, Singapore, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  27. D. Cafagna and G. Grassi, “An effective method for detecting chaos in fractional-order systems,” International Journal of Bifurcation and Chaos, vol. 20, no. 3, pp. 669–678, 2010. View at Publisher · View at Google Scholar · View at Scopus
  28. R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, Springer, Vienna, Austria, 1997. View at Google Scholar · View at MathSciNet
  29. M. Caputo, “Linear models of dissipation whose Q is almost frequency independent. Part II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529–539, 1967. View at Google Scholar
  30. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31–52, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. M. S. Tavazoei and M. Haeri, “A proof for non existence of periodic solutions in time invariant fractional order systems,” Automatica, vol. 45, no. 8, pp. 1886–1890, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. M. S. Tavazoei, “A note on fractional-order derivatives of periodic functions,” Automatica, vol. 46, no. 5, pp. 945–948, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. M. Yazdani and H. Salarieh, “On the existence of periodic solutions in time-invariant fractional order systems,” Automatica, vol. 47, no. 8, pp. 1834–1837, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. R. Caponetto and S. Fazzino, “A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 1, pp. 22–27, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, vol. 16, no. 3, pp. 285–317, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. A. S. Elwakil, “Fractional-order circuits and systems: an emerging interdisciplinary research area,” IEEE Circuits and Systems Magazine, vol. 10, no. 4, pp. 40–50, 2010. View at Publisher · View at Google Scholar · View at Scopus
  38. C.-X. Liu and L. Liu, “Circuit implementation of a new hyperchaos in fractional-order system,” Chinese Physics B, vol. 17, no. 8, pp. 2829–2836, 2008. View at Publisher · View at Google Scholar · View at Scopus
  39. K. Biswas, S. Sen, and P. K. Dutta, “Modeling of a capacitive probe in a polarizable medium,” Sensors and Actuators A, vol. 120, no. 1, pp. 115–122, 2005. View at Publisher · View at Google Scholar · View at Scopus
  40. I. S. Jesus and J. A. Machado, “Development of fractional order capacitors based on electrolyte processes,” Nonlinear Dynamics, vol. 56, no. 1-2, pp. 45–55, 2009. View at Publisher · View at Google Scholar · View at Scopus
  41. D. Zhu, C. X. Liu, and B. Yan, “Drive-response synchronization of a fractional-order hyperchaotic system and its circuit implementation,” Mathematical Problems in Engineering, vol. 2013, Article ID 815765, 8 pages, 2013. View at Publisher · View at Google Scholar