Reduction of Hopf bifurcation problems with symmetries
Here, the Hopf bifurcation is studied in the following family of \(\Gamma\)-invariant differential equations: \(\dot x=f(x,\lambda)\), where \(f: \mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}^n\) is of class \(C^\infty\), \(f(0,\lambda)=0\) for all \(\lambda\) and the matrix \(A=D_1f(0,0)\) has nonzero eigenvalues of the form \(i\mu\) for some integers \(\mu\in Z\). Further \(\Gamma\subset 0(n)\). This bifurcation problem is formulated as an operator equation in spaces of \(2\pi\)-periodic functions \(F(x, \sigma,\lambda)=0\) for \((x,\sigma,\lambda)\) close to \((0,1,0)\) with \(F:C^1_{2\pi}\times \mathbb{R}^1\times \mathbb{R}^m\to C^0_{2\pi}\), with \(F(x,\sigma,\lambda)(t)=-\sigma\dot x(t)+f(x(t),\lambda)\). The paper is well written and organized.
scientific article; zbMATH DE number 1512044