Open AccessDOUBLE HOPF BIFURCATION AND CHAOS IN LIU SYSTEM WITH DELAYED FEEDBACK
Author(s) -
Yuting Ding,
Weihua Jiang
Publication year - 2011
Publication title -
journal of applied analysis and computation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.55
H-Index - 21
eISSN - 2158-5644
pISSN - 2156-907X
DOI - 10.11948/2011023
Subject(s) - bifurcation diagram , hopf bifurcation , mathematics , saddle node bifurcation , bogdanov–takens bifurcation , period doubling bifurcation , pitchfork bifurcation , transcritical bifurcation , center manifold , biological applications of bifurcation theory , bifurcation , mathematical analysis , control theory (sociology) , physics , nonlinear system , computer science , control (management) , quantum mechanics , artificial intelligence
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 bifurcationusing the method of bifurcation analysis. When we choose appropriate feedback strength and delay, two symmetrical nontrivial equilibria of Liusystem 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''.
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