The biharmonic supercritical equation ∆u = |u|p−1u, where n > 4 and p > (n + 4)/(n − 4), is studied in the whole space R as well as in a modified form with λ(1 + u) as right-hand-side with an additional eigenvalue parameter λ > 0 in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the expl...

We give a new characterisation of resonance in Hopf bifurcation from relative equilibria in systems with compact symmetry group. This characterisation provides a full geometric explanation of the resonance phenomenon. In addition, we develop techniques based on normal form theory to give a complete solution to the associated bifurcation problem.

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a ...

We study steady-state bifurcation in reversible equivariant vector fields. We assume an action on the phase space of a compact Lie group G with a normal subgroup H of index two, and study vector fields that are H-equivariant and have all elements of the complement G \ H as time-reversal symmetries. We focus on separable bifurcation problems that can be reduced to equivariant steady-state bifurc...

This pdf document provides the textual background in the mini course on bifurcation analysis, by George van Voorn. It is the companion material by the powerpoint presentations posted on the site www.bio.vu.nl/thb. Goal of this manuscript is to explain some of the basics of bifurcation theory to PhD-students at the group. The emphasis is strongly on the biological interpretation of bifurcations;...

We are concerned with an activator-inhibitor system on an n-dimensional ball such that the inhibitor is activated by an activator as well as the spatial average of its inhibitor. We analyze the existence of the radially symmetric solutions and the occurrence of Hopf bifurcation in the interfacial problem as the bifurcation parameters vary. Mathematics Subject Classification: 35R35, 35B32, 35B25...

In this paper, flip bifurcations of homoclinic orbits in conservative reversible systems are analysed. In such systems, orbit-flip and inclination-flip bifurcations occur simultaneously. It is shown that multi-pulses either do not bifurcate at all at flip bifurcation points or else bifurcate simultaneously to both sides of the bifurcation point. An application to a fifth-order model of water wa...

where g: C?’ --) )? is a given function, Ni are integers and a is a real parameter. Such equations have been considered in several papers, for instance [ 2, 5, 6). It has become apparent that it is important to be able to show that (0.1) has no nontrivial periodic solutions of certain periods. For example, the global Hopf bifurcation theorem, as developed by Alexander and Yorke for ordinary dif...

A method to determine the static bifurcation diagram (sbd) and dynamical bifurcation diagram (dbd) is proposed and the results for a few concrete problems are shown.