Abstract and Applied Analysis

A New Fractional-Order Chaotic Complex System and Its Antisynchronization

Cuimei Jiang, Shutang Liu, and Chao Luo

Full-text: Open access

Abstract

We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 326354, 12 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425049068

Digital Object Identifier
doi:10.1155/2014/326354

Mathematical Reviews number (MathSciNet)
MR3273908

Zentralblatt MATH identifier
07022175

Citation

Jiang, Cuimei; Liu, Shutang; Luo, Chao. A New Fractional-Order Chaotic Complex System and Its Antisynchronization. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 326354, 12 pages. doi:10.1155/2014/326354. https://projecteuclid.org/euclid.aaa/1425049068


Export citation

References

  • 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: Fundamental Theory and Applications, vol. 42, no. 8, pp. 485–490, 1995.
  • C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A: Statistical Mechanics and its Applications, vol. 341, no. 1–4, pp. 55–61, 2004.
  • I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review Letters, vol. 91, no. 3, Article ID 034101, 2003.
  • C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons and Fractals, vol. 22, no. 3, pp. 549–554, 2004.
  • J. G. Lu, “Chaotic dynamics of the fractional-order Lü system and its synchronization,” Physics Letters A, vol. 354, no. 4, pp. 305–311, 2006.
  • L. Pan, W. Zhou, J. Fang, and D. Li, “Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3754–3762, 2010.
  • G. Si, Z. Sun, Y. Zhang, and W. Chen, “Projective synchronization of different fractional-order chaotic systems with non-identical orders,” Nonlinear Analysis: Real World Applications, vol. 13, no. 4, pp. 1761–1771, 2012.
  • L. Chen, Y. Chai, and R. Wu, “Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems,” Physics Letters A: General, Atomic and Solid State Physics, vol. 375, no. 21, pp. 2099–2110, 2011.
  • E. Roldán, G. J. de Valcárcel, R. Vilaseca, and P. Mandel, “Single-mode-laser phase dynamics,” Physical Review A, vol. 48, no. 1, pp. 591–598, 1993.
  • C.-Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Physical Review A, vol. 41, no. 7, pp. 3826–3837, 1990.
  • V. Y. Toronov and V. L. Derbov, “Boundedness of attractors in the complex Lorenz model,” Physical Review E, vol. 55, no. 3, pp. 3689–3692, 1997.
  • G. M. Mahmoud and T. Bountis, “The dynamics of systems of complex nonlinear oscillators: a review,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 11, pp. 3821–3846, 2004.
  • E. E. Mahmoud, “Complex complete synchronization of two nonidentical hyperchaotic complex nonlinear systems,” Mathematical Methods in the Applied Sciences, vol. 37, no. 3, pp. 321–328, 2014.
  • Z. Wu, J. Duan, and X. Fu, “Complex projective synchronization in coupled chaotic complex dynamical systems,” Nonlinear Dynamics, vol. 69, no. 3, pp. 771–779, 2012.
  • F. Zhang and S. Liu, “Full state hybrid projective synchronization and parameters identification for uncertain chaotic (hyperchaotic) complex systems,” Journal of Computational and Nonlinear Dynamics, vol. 9, no. 2, Article ID 021009, 2014.
  • G. M. Mahmoud and E. E. Mahmoud, “Complex modified projective synchronization of two chaotic complex nonlinear systems,” Nonlinear Dynamics, vol. 73, no. 4, pp. 2231–2240, 2013.
  • S. Liu and F. Zhang, “Complex function projective synchronization of complex chaotic system and its applications in secure communication,” Nonlinear Dynamics, vol. 76, no. 2, pp. 1087–1097, 2014.
  • C. Luo and X. Wang, “Chaos in the fractional-order complex Lorenz system and its synchronization,” Nonlinear Dynamics, vol. 71, no. 1-2, pp. 241–257, 2013.
  • C. Luo and X. Wang, “Chaos generated from the fractional-order complex Chen system and its application to digital secure communication,” International Journal of Modern Physics C, vol. 24, Article ID 1350025, 23 pages, 2013.
  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, San Diego, Calif, USA, 1974.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  • G. M. Mahmoud, T. Bountis, and E. E. Mahmoud, “Active control and global synchronization of the complex Chen and Lü systems,” International Journal of Bifurcation and Chaos, vol. 17, no. 12, pp. 4295–4308, 2007.
  • 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.
  • X. Wang, Y. He, and M. Wang, “Chaos control of a fractional order modified coupled dynamos system,” Nonlinear Analysis: Theory, Methods &; Applications, vol. 71, no. 12, pp. 6126–6134, 2009.
  • A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp. 285–317, 1985.
  • P. M. Pardalos and V. A. Yatsenko, “Optimization approach to the estimation and control of Lyapunov exponents,” Journal of Optimization Theory and Its Applications, vol. 128, no. 1, pp. 29–48, 2006.
  • S. P. Nair, D.-S. Shiau, J. C. Principe et al., “An investigation of EEG dynamics in an animal model of temporal lobe epilepsy using the maximum Lyapunov exponent,” Experimental Neurology, vol. 216, no. 1, pp. 115–121, 2009.
  • D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the International IMACS IEEE-SMC Multiconference on Computational Engineering in Systems Applications, Lille, France, 1996. \endinput