DOUBLE HOPF BIFURCATION FOR STUART–LANDAU SYSTEM WITH NONLINEAR DELAY FEEDBACK AND DELAY-DEPENDENT PARAMETERS

Suqi Ma (), Qishao Lu () and S. John Hogan ()
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Suqi Ma: School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, P. R. China
Qishao Lu: School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, P. R. China
S. John Hogan: Department of Engineering Mathematics, University of Bristol, BS8 1TR, Bristol, United Kingdom

Advances in Complex Systems (ACS), 2007, vol. 10, issue 04, 423-448

Abstract: A Stuart–Landau system under delay feedback control with the nonlinear delay-dependent parametere-pτis investigated. A geometrical demonstration method combined with theoretical analysis is developed so as to effectively solve the characteristic equation. Multi-stable regions are separated from unstable regions by allocations of Hopf bifurcation curves in (p,τ) plane. Some weak resonant and non-resonant oscillation phenomena induced by double Hopf bifurcation are discovered. The normal form for double Hopf bifurcation is deduced. The local dynamical behavior near double Hopf bifurcation points are also clarified in detail by using the center manifold method. Some states of two coexisting stable periodic solutions are verified, and some torus-broken procedures are also traced.

Keywords: Stuart–Landau system; double Hopf bifurcation; time delay; feedback control (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1142/S0219525907001227

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