Uniform persistence and Hopf bifurcations in \(\mathbb R_+^n\) (Q2441679)

Consider a parametrized family of uniformly dissipative flows on a locally compact metrizable space and assume uniform persistence at time 0. This article proves that the uniform persistence continues uniformly at 0 if and only if the elements of a Morse decomposition of the maximal compact invariant set of the flow at time 0 are all strongly isolated. In the second part of the paper the authors study generalized Poincaré-Andronov-Hopf bifurcations of parametrized families of flows and derive results regarding the shape of attractors. Finally, conditions are given under which a flow with extreme non-permanence becomes uniformly persistent after a Poincaré-Andronov-Hopf bifurcation.