A new approach for the computation of Hopf bifurcation points (Q5932114)

Considering an \(n\)-dimensional differential system \(\dot x+ f(x,\lambda)= 0\) with real parameter \(\lambda\), a new numerical approach for the computation of Hopf bifurcation points \((x_0,\lambda_0)\) is proposed. With this technique, one only has to solve a system of \(n+2\) equations for \(x_0\), \(\lambda_0\), and the imaginary eigenvalue of \(Df(x_0,\lambda_0)\). Former techniques (see Roose and Hlavaček) required the solution to a system of equations in the higher-dimensional \((2n+2)\)-space. For comparison, the author computes the Hopf bifurcation point for a Brusselator reaction model \((n=4)\) using Newton's method. Here, the present method needs four iterations, while the former technique requires five iterations.