Generalized Hopf bifurcation in Hilbert space

摘要

AbstractWe consider a family of semilinear evolution equations in Hilbert space of the formwith, in general, unbounded operators *A(λ),F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e.F(λ,0) = 0, for all λ εRand that the linearization (with respect tou) at the origin is given by du/dt +A(λ)u= 0. Our essential assumption is the following:A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0)>0 andfor somemεNorIfm= 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e.m>1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up tomsuch branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinar