Stability of singular Hopf bifurcations
Consider the singularly perturbed system \[ \frac{dx}{dt}=\varepsilon f(x,y,\lambda),\quad\frac{dy}{dt}= g(x,y,\lambda)\tag{*} \] with \(x\in\mathbb{R}^n\), \(y\in\mathbb{R}^m\), \(\varepsilon,\lambda\in\mathbb{R}\), \(\varepsilon\) small. It is assumed that there exists a curve \(\lambda=\lambda(\varepsilon)\) in the parameter plane associated with the occurence of Hopf bifurcation. The goal of the authors is to derive an explicit formula for the so-called Lyapunov constant \(\sigma\) whose sign determines the stability of the bifurcating small limit cycle. The derivation of this formula is based on a reduction technique. It requires that \(f\) and \(g\) are sufficiently smooth and that the corresponding Jacobi matrix satisfies some spectral conditions.
scientific article; zbMATH DE number 2116217
scientific article