Bifurcations, symmetries and the notion of fixed subspace

Abstract

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

Rotational Symmetry and Rotating Waves in Planar Integro-Difference Equations

“…and claim that F ′ (t) < 0 for all t > 0, from which it follows that F (t) ≥ 1 for all t > 0 from the above asymptotic behaviour of F as t → 0 + and as t → ∞. To prove the claim, we use formula (10) We note that for every t ≥ 0, we can show using simple calculus arguments that 2 e t 2 /2 ≥ 2 3/4 t 3/2 from which it follows (using t → t…”

“…which is positive because of formula (10) from [35] (see also (A.5) in the Appendix) which allows us to write…”

“…If we then consider the dynamical system on X defined by iterations of the map N in (1.5), equation (1.6) implies that the group SO(2) maps orbits of this dynamical system into other orbits of the dynamical system. The study of group equivariant dynamical systems has enjoyed a rich development over the past decades, see for example [1,8,9,10,13,14,16,23]. An interesting class of orbits which may exist for SO(2)-equivariant dynamical systems are so-called rotating waves.…”