Hopf bifurcation from infinity
The following result is proved: ``Let \(\{\pi_{\mu}\}_{\mu}\) be a one- parameter continuous family of flows on X, a locally compact metric space. If \(\pi_ 0\) is negatively ultimately bounded, and for \(\mu >0\pi_{\mu}\) is ultimately bounded, then there is a family of compact, invariant, asymptotically stable sets \(M_{\mu}\) bifurcating from infinity'', If \(X={\mathbb{R}}^ 2 \)and all the fixed points of \(\pi_{\mu}\) are contained in a disk independent of \(\mu\), then \(M_{\mu}\) contains a cycle, for \(\mu\) positive. The above result is applied to prove the existence of periodic solutions bifurcating from infinity for second order differential equations of generalized Liénard and Rayleigh type.
scientific article; zbMATH DE number 4050050
scientific article