Nonlinear Dynamics

, Volume 39, Issue 4, pp 319–334

Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems

  • Tianshou Zhou
  • Guanrong Chen
  • Sergej ČelikovskÝ

DOI: 10.1007/s11071-005-4195-8

Cite this article as:
Zhou, T., Chen, G. & ČelikovskÝ, S. Nonlinear Dyn (2005) 39: 319. doi:10.1007/s11071-005-4195-8


This paper studies the generalized Lorenz canonical form of dynamical systems introduced by Čelikovský and Chen [International Journal of Bifurcation and Chaos12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The Ši’lnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have Ši’lnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of Ši’lnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.

Key words

heteroclinic orbit generalized Lorenz canonical form Ši’lnikov criterion 

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Tianshou Zhou
    • 1
  • Guanrong Chen
    • 2
  • Sergej ČelikovskÝ
    • 3
  1. 1.Department of MathematicsZhongshan UniversityGuangzhouP.R. China
  2. 2.Department of Electronic EngineeringCity University of Hong KongP.R. China
  3. 3.Department of Control Engineering, Faculty of Electrical EngineeringCzech Technical UniversityPragueCzech Republic

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