Relaxation oscillations in \({\mathbb R}^3\)
This paper is concerned with relaxation oscillations which arise from the singularly perturbed system \(\dot{x}=g_{1}(x,y,z,\varepsilon )\), \(\dot{y}=g_{2}(x,y,z,\varepsilon )\), \(\varepsilon \dot{z}=f(x,y,z,\varepsilon )\) with \(0< \varepsilon \ll 1\). The system has two slow and one fast variable. It is proved, under certain restrictions, that there exists a periodic relaxation orbit for small \(\varepsilon\). Proofs use geometric singular perturbation theory and blow-up methods. A singular form of the forced van der Pol equation is discussed in some detail.
scientific article; zbMATH DE number 2104506