Singular heteroclinic cycles
The paper presents a study of bifurcations from singular heteroclinic cycles in two parameter families of vector fields in \(\mathbb{R}^n\). The heteroclinic cycle consists of a hyperbolic singularity, a saddle-node singularity, and two heteroclinic orbits between them. The weak stable and weak unstable eigenvalues of the vector field at the hyperbolic singularity are assumed to be real. Whereas the weak hyperbolic eigenvalues at the saddle-node may be either real or complex conjugate. The last case is of particular interest since, as shown in the paper, under some eigenvalue conditions a complex bifurcation structure (a cascade of inclination flip bifurcations) leading to a chaotic dynamics can appear. Bifurcations from those of the cycles that form a codimension two phenomenon, i.e. they occur persistently in two parameter families at isolated parameter values, are treated. The bifurcation diagrams of all such bifurcations are discussed. For the cases with a simple dynamics in the unfolding, the complete bifurcation diagrams are obtained. The bifurcation analysis is based on various techniques: the implicit mapping theorem, a Lyapunov-Schmidt reduction, Shil'nikov variables, the construction of invariant manifolds and foliations by using graph transforms.
scientific article; zbMATH DE number 1437332
scientific article