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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 186037, 6 pages
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.
- E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of The Atmospheric Sciences, vol. 20, pp. 130–141, 1963.
- 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.
- 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.
- 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.
- 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.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- 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.
- J. A. T. Machado, “Entropy analysis of integer and fractional dynamical systems,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 371–378, 2010.
- 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.
- 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.
- 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.
- 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.
- 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.
- V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1117–1127, 2010.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, New York, NY, USA, 1993.
- 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.
- Z. Vukic, Lj. Kuljaca, D. Donlagic, and S. Tesnjak, Non-linear Control Systems, Marcel Dekker, New York, NY, USA, 2003.
- 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.
- M. S. Tavazoei and M. Haeri, “Chaotic attractors in incommensurate fractional order systems,” Physica D, vol. 237, no. 20, pp. 2628–2637, 2008.
- 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.
- 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.
- 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.
- C. P. Silva, “Shilnikov theorem—a tutorial,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 10, pp. 675–682, 1993.
- 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.
- 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.
- 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.
- 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.