Hopf bifurcation with constraints

Abstract

IN THIS study we give conditions for the existence of periodic orbits of a strongly continuous semiflow which branch off from a steady state for a certain parameter value. The semiflow is defined in a closed subset of a Banach space which contains the steady state. The steady state is kept fixed whereas both the semiflow and its domain may depend strongly continuously on the parameter. It will be shown that an asymptotic @-contraction hypothesis together with a transversality and no-resonance hypothesis for the eigenvalues of the infinitesimal generator of the linearized semiflow imply Hopf bifurcation. The technique consists in a generalized Lyapunov-Schmidt decomposition method using spectral projections which depend strongly continuously on the bifurcation parameter. The existence of a continuum of bifurcating periodic orbits follows from a degree argument. The theory can be applied to partial functional differential equations when the bifurcation parameter changes transformed arguments or the initial or boundary values underly parameter-dependent constraints. In general such semiflows do not exhibit the smoothness hypotheses which are necessary for the application of available Hopf bifurcation theorems. There are. a lot of such theorems either for special types of functional differential equations, partial differential equations, Volterra-integral equations, or for a general semiflow in Banach space but under considerable stronger hypotheses than in the present paper. The reader is referred to [l], [2], [4], [5], [7]-[12], [15] and to the references therein. Of course the stronger hypotheses in these papers allow to get more detailed information on the structure of the branch of the bifurcating periodic orbits such as smootheness with respect to the parameter, local stability, and the direction of bifurcation. In contrast to them the hypotheses in our paper guarantee only local existence and they are too weak to ensure smoothness of the branch nor the existence of a center manifold. But even the local existence problem was still unsolved if the semiflow and its domain depend merely strongly continuously on the parameter. The difficulty is due to the fact that the usual argument basing on the implicit function theorem will not work. Some examples which demonstrate the application of our theory will be given in a subsequent paper. Before we will explain the main hypotheses we shall introduce some notations. B denotes a real Banach space and C C Iw’ a compact neighborhood of 0 E R*, the domain of the bifurcation-parameter. Norms and the modulus of a complex number are always indicated by the symbol 1.1. To every c E C we join a closed subset X’ C B, the domain of the semiflow.