On the foundations of bifurcation theory

On Bifurcations Arising from Unstable Equilibria and Invariant Sets

“…Just as in the case of the AS CI bifurcations (and their generalization in [7]), the fundamental underlying principle was the persistence property of asymptotic stability, in the present case the underlying principle is another persistence property, namely the persistence of instability. By this we mean that the extent of instability of M, measured by its positive prolongation (in the sense of Ura), cannot decrease discontinuously, though it can increase discontinuously.…”

“…In the theory developed in [5] and [7], the fundamental underlying principle was that of persistence of asymptotic stability under perturbations, a consequence of which is the relation between extracritical loss of stability (defined above) and bifurcation.…”

“…In what follows, we will refer to the class of bifurcations considered in [5] as AS CI (asymptotic stability complete instability) bifurcations. In the paper [7] by the present authors, the requirement of [5] that M be completely unstable was weakened, and the theory was extended to the case of semidynamical systems, dropping the requirement of local compactness of the state space and assuming instead that the systems be asymptotically compact.…”

On certain generalizations of the Hopf bifurcation

“…In [13], on the other hand, the condition of complete instability for extracritical values of λ (i.e. λ = λ 0 ) was replaced by various weaker conditions, one of which was that M should not be an attractor or weak attractor.…”

“…These conditions are also satisfied in the classical Hopf case (actually, for two-dimensional systems, to which the higher dimensional cases can be reduced by centre manifold techniques), but they need not be reflected in the linear parts of the systems. The principal methodological tool used in [6], and in the later paper [13], was a persistence principle of asymptotic stability: if a system with a stable attractor is subjected to a sufficiently small perturbation, the new system exhibits a stable attractor in an arbitrarily small neighbourhood of the original attractor.…”