Abstract Hopf bifurcation theorem and further extensions via second variation
The paper is devoted to the bifurcation problem in Banach spaces of the form \(F(\mu, x)=0\), where \(F\) is a \(C^2\) map and \(\frac{\partial F}{\partial x}(\mu _0, x_0)\) is a Fredholm operator of index \(0\). While the abstract results are mainly cited from the earlier article [\textit{A. V. Arutyunov, A. F. Izmailov} and \textit{V. Jaćimović}, ``New bifurcation theorems via the second-order optimality conditions'', J. Math. Anal. Appl. 359, No.~2, 752--764 (2009; Zbl 1168.47050)], the author considers especially one corollary and gives three examples of its applications in ordinary and partial differential problems. More precisely, he investigates a special type of bifurcation and assumes that the parameter space and the kernel of \(\frac{\partial F}{\partial x}(\mu _0, x_0)\) have the same (finite) dimensions. As he proves, such methods are possible only when these dimensions are equal to \(1\), \(2\), \(4\) or \(8\), which is related to the Hopf fibrations of spheres.
scientific article; zbMATH DE number 5774957