Bifurcation of periodic solutions from inversion of stability of periodic O.D.E.'s (Q1313418)

A parameterized family of differential equations \(x'= f(\lambda,t,x)\), \(f\) \(T\)-periodic in \(t\), where \(\lambda\in [0,\lambda_ 0)\) and \(x\in \mathbb{R}^ n\) is considered. The main result states that if \(n\) is odd, \(\{u_ \lambda\}\) is a family of \(T\)-periodic solutions continuous with respect to \(\lambda\), \(u_ 0\) is asymptotically stable and \(u_ \lambda\) is negatively asymptotically stable for \(\lambda>0\), then \(\lambda=0\) is a bifurcation parameter for \(\{u_ \lambda\}\). The result is a consequence of the Lefschetz Fixed Point theorem applied to the corresponding Poincaré operator restricted to the set bounded by two concentric spheres.