Topological Bifurcation

Symmetric Liapunov center theorem

“…Finally, they apply the Morse theory or the Conley index theory to prove the existence of bifurcation of periodic solutions. In this way they obtain a local bifurcation (a sequence of solutions bifurcating from the family of trivial ones) which does not have to be global (a connected set of solutions bifurcating from the family of trivial ones), see [3,10,23,26,36] for discussions and examples.…”

Equivariant bifurcation index

“…This theorem yields information about bifurcations of connected sets of orbits of critical points of G-invariant functionals. The proof of this theorem follows very closely the proof of the classical Rabinowitz alternative, see [18,[41][42][43][44][45], with the Leray-Schauder degree replaced by the degree for G-equivariant gradient maps. (2) or the continuum C ν −1 k 0 ⊂ H × R is bounded, and additionally the following conditions are satisfied…”

A short treatise on the equivariant degree theory and its applications

“…The equivariant degree theory, which emerged in nonlinear analysis about 20 years ago, has many natural roots: Borsuk-Ulam theorems (see [98,72,60]), fundamental domains and equivariant retract theory (see [72,2,13,60]), equivariant homotopy groups of spheres (see [73,33,60,13,75]), equivariant bordism theory (see [99,13]), equivariant general position theorems (see [72,46,60]), gradient S 1 -invariants (see [28,31]), Whitehead J-homomorphism (see [56]). Two monographs on this subject (see [13,60]) can provide an experienced reader with a systematic exposition of the equivariant degree theory in all its aspects (see also the survey papers [57,7,96]).…”

“…For each k = 1, 2, 3, take β = β k defined by (6.18) (with ξ replaced by ξ k ) and r 0 = r k defined by (6.19) (with ξ replaced by ξ k and m = 0). Given a pair (r k , β k ) and using the standard auxiliary function techniques (see [56] or [13]), one can establish…”