Abstract and Applied Analysis

Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System

Ranchao Wu and Xiang Li

Full-text: Open access


A new Rössler-like system is constructed by the linear feedback control scheme in this paper. As well, it exhibits complex dynamical behaviors, such as bifurcation, chaos, and strange attractor. By virtue of the normal form theory, its Hopf bifurcation and stability are investigated in detail. Consequently, the stable periodic orbits are bifurcated. Furthermore, the anticontrol of Hopf circles is achieved between the new Rössler-like system and the original Rössler one via a modified projective synchronization scheme. As a result, a stable Hopf circle is created in the controlled Rössler system. The corresponding numerical simulations are presented, which agree with the theoretical analysis.

Article information

Abstr. Appl. Anal., Volume 2012 (2012), Article ID 341870, 16 pages.

First available in Project Euclid: 14 December 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Wu, Ranchao; Li, Xiang. Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System. Abstr. Appl. Anal. 2012 (2012), Article ID 341870, 16 pages. doi:10.1155/2012/341870. https://projecteuclid.org/euclid.aaa/1355495792

Export citation


  • E. N. Lorenz, “Deterministic non-periodic flows,” Journal of Atomic and Molecular Sciences, vol. 20, pp. 130–141, 1963.
  • O. E. R. Rössler, “Continuous chaosfour prototype equations,” Annals of the New York Academy of Sciences, vol. 316, pp. 376–392, 1979.
  • X. Liao and P. Yu, “Study of globally exponential synchronization for the family of Rössler systems,” International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2395–2406, 2006.
  • Y. Liu and Q. Yang, “Dynamics of a new Lorenz-like chaotic system,” Nonlinear Analysis, vol. 11, no. 4, pp. 2563–2572, 2010.
  • G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos, vol. 9, no. 7, pp. 1465–1466, 1999.
  • J. Lü, G. Chen, and S. Zhang, “Dynamical analysis of a new chaotic attractor,” International Journal of Bifurcation and Chaos, vol. 12, no. 5, pp. 1001–1015, 2002.
  • X. Zhou, Y. Wu, Y. Li, and Z. X. Wei, “Hopf bifurcation analysis of the Liu system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1385–1391, 2008.
  • R. A. Van Gorder and S. R. Choudhury, “Analytical Hopf bifurcation and stability analysis of T system,” Communications in Theoretical Physics, vol. 55, no. 4, pp. 609–616, 2011.
  • Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112, Springer, New York, NY, USA, 1998.
  • F. Dercole and S. Maggi, “Detection and continuation of a border collision bifurcation in a forest fire model,” Applied Mathematics and Computation, vol. 168, no. 1, pp. 623–635, 2005.
  • Y. Ma, M. Agarwal, and S. Banerjee, “Border collision bifurcations in a soft impact system,” Physics Letters A, vol. 354, no. 4, pp. 281–287, 2006.
  • J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, NY, USA, 1983.
  • C. C. Yu, Autotuning of PID Controllers: A Relay Feedback Approach, Springer, Berlin, Germany, 1999.
  • Q. G. Wang, T. H. Lee, and C. Lin, Relay Feedback: Analysis, Identification and Control, Springer, London, UK, 2003.
  • E. H. Abed and J.-H. Fu, “Local feedback stabilization and bifurcation control. I. Hopf bifurcation,” Systems & Control Letters, vol. 7, no. 1, pp. 11–17, 1986.
  • P. K. Yuen and H. H. Bau, “Rendering a subcritical Hopf bifurcation supercritical,” Journal of Fluid Mechanics, vol. 317, pp. 91–109, 1996.
  • G.-H. Li, “Modified projective synchronization of chaotic system,” Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1786–1790, 2007.
  • B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, Mass, USA, 1982.
  • G. Wen, “Designing Hopf limit circle to dynamical systems via modified projective synchronization,” Nonlinear Dynamics, vol. 63, no. 3, pp. 387–393, 2011.
  • P. N. Paraskevopoulos, Morden Control Engingneering, Marcel Dekker, New York, NY, USA, 2002.