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Volume 20,     Number 2,     Summer 2012

 

CENTER-UNSTABLE MANIFOLDS FOR
NONDENSELY DEFINED CAUCHY PROBLEMS
AND APPLICATIONS TO STABILITY OF HOPF
BIFURCATION

Dedicated to Professor Herbert I. Freedman on the occasion of his 70th birthday.

ZHIHUA LIU, PIERRE MAGAL AND SHIGUI RUAN

Abstract. Center-unstable manifolds are very useful in investigating nonlinear dynamics of nonlinear evolution equations. In this paper, we rst present a center-unstable manifold theory for abstract semilinear Cauchy problems with nondense domain. We especially focus on the stability property of the center-unstable manifold. Then we study the stability of Hopf bifurcation, that is, stability of the bifurcating periodic orbits for the nondensely de ned Cauchy problem. Our goal is to prove that the stability of a periodic orbit to the reduced system (i.e., restricted to the center-unstable manifold) implies the stability of the periodic orbit for the original system. As an application, we demonstrate that these results apply to differential equations with in nite delay.

 

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