Abstract and Applied Analysis

Stability and Stabilizing of Fractional Complex Lorenz Systems

Rabha W. Ibrahim

Full-text: Open access

Abstract

We study the stability and stabilization of complex fractional Lorenz system. The fractional calculus are taken in sense of the Caputo derivatives. The technique is based on stability theory of fractional-order systems. Numerical solutions are imposed.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 127103, 13 pages.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1366306770

Digital Object Identifier
doi:10.1155/2013/127103

Mathematical Reviews number (MathSciNet)
MR3035198

Zentralblatt MATH identifier
1267.34072

Citation

Ibrahim, Rabha W. Stability and Stabilizing of Fractional Complex Lorenz Systems. Abstr. Appl. Anal. 2013 (2013), Article ID 127103, 13 pages. doi:10.1155/2013/127103. https://projecteuclid.org/euclid.aaa/1366306770


Export citation

References

  • E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, vol. 20, no. 2, pp. 130–1341, 1963.
  • V. Afraimovic, V. Bykov, and L. P. Shilnikov, “Origin and structure of the Lorenz attractor,” Soviet Physics–-Doklady, vol. 22, pp. 253–255, 1977.
  • C. Sparrow, The Lorenz equations: bifurcations, chaos, and strange attractors, vol. 41 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1982.
  • M. F. Doherty and J. M. Ottino, “Chaos in deterministic systems: strange attractors, turbulence, and applications in chemical engineering,” Chemical Engineering Science, vol. 43, no. 2, pp. 139–183, 1988.
  • K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos, Textbooks in Mathematical Sciences, Springer, New York, NY, USA, 1997.
  • J. C. Sprott, “Some simple chaotic flows,” Physical Review E, vol. 50, no. 2, pp. R647–R650, 1994.
  • 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.
  • A. Vaněček and S. Čelikovský, “Bilinear systems and chaos,” Kybernetika, vol. 30, no. 4, pp. 403–424, 1994.
  • S. Vaněček and A. Čelikovský, Control Systems: From Linear Analysis to Synthesis of Chaos, Prentice-Hall, London, UK, 1996.
  • J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 3, pp. 659–661, 2002.
  • S. Čelikovský and G. Chen, “On the generalized Lorenz canonical form,” Chaos, Solitons and Fractals, vol. 26, no. 5, pp. 1271–1276, 2005.
  • Q. Yang, G. Chen, and T. Zhou, “A unified Lorenz-type system and its canonical form,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 10, pp. 2855–2871, 2006.
  • Q. Yang, G. Chen, and K. Huang, “Chaotic attractors of the conjugate Lorenz-type system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 17, no. 11, pp. 3929–3949, 2007.
  • K. Huang and Q. Yang, “Stability and Hopf bifurcation analysis of a new system,” Chaos, Solitons and Fractals, vol. 39, no. 2, pp. 567–578, 2009.
  • Q. Zhang, J. H. L. Lü, and S. H. Chen, “Coexistence of anti-phase and complete synchronization in the generalized Lorenz system,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 10, pp. 3067–3072, 2010.
  • X. Shi and Z. Wang, “The alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay,” Nonlinear Dynamics, vol. 69, no. 3, pp. 1177–1190, 2012.
  • S. Camargo, L. R. L. Viana, and C. Anteneodo, “Intermingled basins in coupled Lorenz systems,” Physical Review E, vol. 85, no. 3, Article ID 036207, 10 pages, 2012.
  • S. Li, Y. Li, B. Liu, and T. Murray, “Model-free control of Lorenz chaos using an approximate optimal control strategy,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4891–4900, 2012.
  • S. Čelikovský and G. Chen, “On a generalized Lorenz canonical form of chaotic systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 8, pp. 1789–1812, 2002.
  • S. Čelikovský and G. Chen, “Hyperbolic-type generalized Lorenz system and its canonical form,” in Proceedings of the 15th Triennial World Congress of IFAC, Barcelona, Spain, July 2002.
  • I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional order Lorenz system,” Physical Review Letters, vol. 91, no. 3, Article ID 03410, 4 pages, 2003.
  • C. P. Li and G. J. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 443–450, 2004.
  • J. P. Yan and C. P. Li, “On chaos synchronization of fractional differential equations,” Chaos, Solitons and Fractals, vol. 32, no. 2, pp. 725–735, 2007.
  • P. Zhou P and X. Cheng, “Synchronization between different fractional order chaotic systems,” in Proceeding of the 7th World Congress on Intelligent Control and Automation, Chongqing, China, June 2008.
  • Y. Yu, H. X. Li, S. Wang, and J. Yu, “Dynamic analysis of a fractional-order Lorenz chaotic system,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1181–1189, 2009.
  • K. Sun and J. C. Sprott, “Bifurcations of fractional-order diffusionless lorenz system,” Electronic Journal of Theoretical Physics, vol. 6, no. 22, pp. 123–134, 2009.
  • S. K. Agrawal, M. Srivastava, and S. Das, “Synchronization of fractional order chaotic systems using active control method,” Chaos, Solitons and Fractals, vol. 45, no. 6, pp. 737–752, 2012.
  • Y. Xu, R. Gu, H. Zhang, and D. Li, “Chaos in diffusionless Lorenz system with a fractional order and its control,” International Journal of Bifurcation and Chaos, vol. 22, no. 4, pp. 1–8, 2012.
  • P. Zhou and R. Ding, “Control and synchronization of the fractional-order Lorenz chaotic system via fractionalorder derivative,” Mathematical Problems in Engineering, vol. 2012, Article ID 214169, 14 pages, 2012.
  • G. Si, Z. Sun, H. Zhang, and Y. Zhang, “Parameter estimation and topology identification of uncertain fractional order complex networks,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 5158–5171, 2012.
  • A. C. Fowler, M. J. McGuinness, and J. D. Gibbon, “The complex Lorenz equations,” Physica D, vol. 4, no. 2, pp. 139–163, 1981/82.
  • A. C. Fowler, J. D. Gibbon, and M. J. McGuinness, “The real and complex Lorenz equations and their relevance to physical systems,” Physica D, vol. 7, no. 1–3, pp. 126–134, 1983.
  • 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.
  • A. D. Kiselev, “Symmetry breaking and bifurcations in complex Lorenz model,” Journal of Physical Studies, vol. 2, no. 1, pp. 30–37, 1998.
  • A. Rauh, L. Hannibal, and N. B. Abraham, “Global stability properties of the complex Lorenz model,” Physica D, vol. 99, no. 1, pp. 45–58, 1996.
  • S. Panchev and N. K. Vitanov, “On asymptotic properties of some complex Lorenz-like systems,” Journal of the Calcutta Mathematical Society, vol. 1, no. 3-4, pp. 121–130, 2005.
  • G. M. Mahmoud, M. A. Al-Kashif, and S. A. Aly, “Basic properties and chaotic synchronization of complex Lorenz system,” International Journal of Modern Physics C, vol. 18, no. 2, pp. 253–265, 2007.
  • G. M. Mahmoud, M. E. Ahmed, and E. E. Mahmoud, “Analysis of hyperchaotic complex Lorenz systems,” International Journal of Modern Physics C, vol. 19, no. 10, pp. 1477–1494, 2008.
  • E. E. Mahmoud and G. M. Mahmoud, Chaotic and Hyperchaotic Nonlinear Systems, Lambert Academic Publishing, Saarbrücken, Germany, 2011.
  • G. M. Mahmoud and E. E. Mahmoud, “Complete synchronization of chaotic complex nonlinear systems with uncertain parameters,” Nonlinear Dynamics, vol. 62, no. 4, pp. 875–882, 2010.
  • Z. Li, Z. Duan, L. Xie, and X. Liu, “Distributed robust control of linear multi-agent systems with parameter uncertainties,” International Journal of Control, vol. 85, no. 8, pp. 1039–1050, 2012.
  • E. E. Mahmoud, “Dynamics and synchronization of new hyperchaotic complex Lorenz system,” Mathematical and Computer Modelling, vol. 55, no. 7-8, pp. 1951–1962, 2012.
  • A. M. A. El-Sayed, E. Ahmed, and H. A. A. El-Saka, “Dynamic properties of the fractional-order logistic equation of complex variables,” Abstract and Applied Analysis, vol. 2012, Article ID 251715, 12 pages, 2012.
  • H. Haken, “Analogy between higher instabilities in fluids and lasers,” Physics Letters A, vol. 53, no. 1, pp. 77–88, 1975.
  • X. Li and W. Chen, “Analytical study on the fractional anomalous diffusion in a half-plane,” Journal of Physics A, vol. 43, no. 49, Article ID 495206, 11 pages, 2010.
  • Y. J. Liang and W. Chen, “A survey on numerical evaluation of Lvy stable distributions and a new MATLAB toolbox,” Signal Processing, vol. 93, no. 1, pp. 242–251, 2013.
  • S. Hu, W. Chen, and X. Gou, “Modal analysis of fractional derivative damping model of frequency-dependent viscoelastic soft matter,” Advances in Vibration Engineering, vol. 10, no. 3, pp. 187–196, 2011.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, London, UK, 1999.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • J. Sabatier, O. P. Agrawal, and J. A. Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
  • L. Chen, Y. Chai, R. Wu, and J. Yang, “Stability and stabilization of a class of nonlinear fractional order system with Caputo derivative,” IEEE Transaction on Circuits and Systems, vol. 59, no. 9, pp. 602–606, 2012.
  • 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.
  • D. Matignon, “Stability result on fractional differential equations with applications to control processing,” in Proceedings of the IMACS-SMC 96, vol. 2, pp. 963–968, 1996.