Asymptotic analysis of perturbed Takens-Bogdanov points (Q1182668)

Consider a parameter dependent equation (1) \(F(u,\lambda,\alpha)=0\) defined by a smooth map \(F: R^ N\times R^ 1\times R^ k\to R^ N\). Takens-Bogdanov points are particular singular solutions \((u^*,\lambda^*,\alpha^*)\) of (1) that are of interest because --- under perturbation --- they may give rise to Hopf bifurcations. The authors use a form of Lyapunov-Schmidt reduction to provide a first-order analysis of singular points in a neighborhood of \((u^*,\lambda^*,\alpha^*)\) under small perturbations of \(\alpha\). The results reproduce, in essence, those of \textit{A. Spence, K. A. Cliffe} and \textit{A. D. Jepson} [J. Comput. Appl. Math. 26, No. 1/2, 125-131 (1989; Zbl 0684.65057)] obtained by different techniques. The latter authors also suggested some practical applications of such an asymptotic analysis. Here a numerical example is given.