On bifurcation of periodic solutions for analytic families of vector fields (Q1337631)

The authors study the problem of bifurcation of a periodic solution from the equilibrium at the origin for an analytic family of vector fields \(F^ \mu\) on \(\mathbb{R}^ 2\). In terms of \(F^ \mu\), two auxiliary mappings \(G\), \(H : \mathbb{R}^ 3 \to \mathbb{R}^ 3\) with \(G(0) = H(0) = 0\) are introduced. Assume the local topological degrees \(\deg (G)\) and \(\deg (H)\) at \(0 \in R^ 3\) are defined. The main purpose of this paper is to prove that if \(\deg (G) = \deg (H) = 1\) and if \(F^ 0\) is internally transversal to every small circle centered at (0,0), then for every sufficiently small parameter \(\mu \neq 0\) there exists a nontrivial periodic solution of the vector field \(F^ \mu\) near the origin (0,0). This result is quite different from the famous Hopf theorem as shown by concrete examples in the paper.