About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics
Volume 2013 (2013), Article ID 186037, 6 pages
http://dx.doi.org/10.1155/2013/186037
Research Article

The Proposed Modified Liu System with Fractional Order

1Department of Physics, Urmia Branch, Islamic Azad University, P.O. Box 969, Oromiyeh, Iran
2Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey
3Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania
4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

Received 27 February 2013; Accepted 15 March 2013

Academic Editor: J. A. Tenreiro Machado

Copyright © 2013 Alireza K. Golmankhaneh et al. 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. E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of The Atmospheric Sciences, vol. 20, pp. 130–141, 1963.
  2. J. Lü, T. Zhou, G. Chen, and S. Zhang, “Local bifurcations of the Chen system,” International Journal of Bifurcation and Chaos, vol. 12, no. 10, pp. 2257–2270, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L.J.M. Kocić, S. Gegovka-Zajkovka, and S. Kostadinova, “On Chua dynamical system,” Applied Mathematics, Informatics & Mechanics, Series A, vol. 2, pp. 53–60, 2010.
  4. T. Stachowiak and T. Okada, “A numerical analysis of chaos in the double pendulum,” Chaos, Solitons and Fractals, vol. 29, no. 2, pp. 417–422, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. W. Xuedi and W. Chen, “Bifurcation analysis and control of the Rossler,” in Proceedings of the 7th International Conference on System Natural Computation (ICNC '11), vol. 3, pp. 1484–1488, IEEE, Shanghai, China, 2011.
  6. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  7. C. M. Ionescu and R. De Keyser, “Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease,” IEEE Transactions on Biomedical Engineering, vol. 56, no. 4, pp. 978–987, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. J. A. T. Machado, “Entropy analysis of integer and fractional dynamical systems,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 371–378, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. M. Lopes, J. A. T. Machado, C. M. A. Pinto, and A. M. S. F. Galhano, “Fractional dynamics and MDS visualization of earthquake phenomena,” Computers and Mathematics with Applications, 2013. View at Publisher · View at Google Scholar
  10. L.-J. Sheu, H.-K. Chen, J.-H. Chen et al., “Chaos in the Newton-Leipnik system with fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 98–103, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review Letters, vol. 91, no. 3, pp. 034101/1–034101/4, 2003. View at Scopus
  12. T. T. Hartley, C. F. Lorenzo, and H. K. 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
  13. C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  14. V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1117–1127, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  16. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, New York, NY, USA, 1993. View at MathSciNet
  17. 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
  18. Z. Vukic, Lj. Kuljaca, D. Donlagic, and S. Tesnjak, Non-linear Control Systems, Marcel Dekker, New York, NY, USA, 2003.
  19. A. Razminia, V.J. Majd, and D. Baleanu, “Chaotic incommensurate fractional order Rössler system: active control and synchronization,” Advances in Difference Equations, article 15, 2011. View at Publisher · View at Google Scholar
  20. M. S. Tavazoei and M. Haeri, “Chaotic attractors in incommensurate fractional order systems,” Physica D, vol. 237, no. 20, pp. 2628–2637, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. W. Deng, C. Li, and J. Lü, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409–416, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. S. Tavazoei and M. Haeri, “A necessary condition for double scroll attractor existence in fractional-order systems,” Physics Letters A, vol. 367, no. 1-2, pp. 102–113, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. L. O. Chua, M. Komuro, and T. Matsumoto, “The double scroll family. I. Rigorous proof of chaos,” IEEE Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1072–1097, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  24. C. P. Silva, “Shilnikov theorem—a tutorial,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 10, pp. 675–682, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  25. D. Cafagna and G. Grassi, “New 3D-scroll attractors in hyperchaotic Chua's circuits forming a ring,” International Journal of Bifurcation and Chaos, vol. 13, no. 10, pp. 2889–2903, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. Lü, G. Chen, X. Yu, and H. Leung, “Design and analysis of multiscroll chaotic attractors from saturated function series,” IEEE Transactions on Circuits and Systems I, vol. 51, no. 12, pp. 2476–2490, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  27. 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
  28. M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D, vol. 65, no. 1-2, pp. 117–134, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet