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Yuting Ding, Weihua Jiang. DOUBLE HOPF BIFURCATION AND CHAOS IN LIU SYSTEM WITH DELAYED FEEDBACK[J]. Journal of Applied Analysis & Computation, 2011, 1(3): 325-349. doi: 10.11948/2011023
Citation: Yuting Ding, Weihua Jiang. DOUBLE HOPF BIFURCATION AND CHAOS IN LIU SYSTEM WITH DELAYED FEEDBACK[J]. Journal of Applied Analysis & Computation, 2011, 1(3): 325-349. doi: 10.11948/2011023
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DOUBLE HOPF BIFURCATION AND CHAOS IN LIU SYSTEM WITH DELAYED FEEDBACK

  • Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

    Abstract
  • In this paper, we consider the stability of equilibria, Hopf and double Hopf bifurcation in Liu system with delay feedback. Firstly, we identify the critical values for stability switches and Hopf bifurcation using the method of bifurcation analysis. When we choose appropriate feedback strength and delay, two symmetrical nontrivial equilibria of Liu system can be controlled to be stable at the same time, and the stable bifurcating periodic solutions occur in the neighborhood of the two equilibria at the same time. Secondly, by applying the normal form method and center manifold theory, the normal form near the double Hopf bifurcation, as well as classifications of local dynamics are analyzed. Furthermore, we give the bifurcation diagram to illustrate numerically that a family of stable periodic solutions bifurcated from Hopf bifurcation occur in a large region of delay and the Liu system with delay can appear the phenomenon of "chaos switchover".
    MSC: 93C10;37G10;34C28;37G05
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