Hopf bifurcation: the appearance of virtual periods in cases of resonance
The paper is concerned with an extension of the Hopf theorem in certain critical cases. The main result proved is the following. Let (0,0) be an isolated centre of (1) \(dx/dt=f(x,\lambda)\), where \(f\in C^ 1(R^ n\times R,R^ n)\), and let \(i\beta\) be an eigenvalue of \(A=D_ xf(0,0)\). If the crossing number \(\chi(i\beta)\neq 0\) and if \(\pi/\beta\) is not a centre period of (0,0), then there exists a family Q of orbits of (1) such that Q bifurcates from (0,0) starting with virtual period \(2\pi/\beta\). Several examples are included.
scientific article; zbMATH DE number 3998836
scientific article