Extending Slow Manifolds to Nonhyperbolic Points in Planar Singularly Perturbed Systems When Linearly Slow Flow Has No Directions

ABSTRACT By employing geometric singular perturbation theory along with the techniques of blow‐up and generalized rotated vector field, this paper aims to extend the slow flow to nonhyperbolic points in planar singularly perturbed systems. We are mainly concerned with a more degenerate situation, where the linearly slow flow has no direction. Under this setting, this paper gives a classification of the flow near transcritical singularity, which is a typical nonhyperbolic point in planar singularly perturbed systems. This more degenerate case gives rise to new dynamical behaviors, such as the slow flow exhibiting opposite direction relative to the reduced flow on critical manifold after perturbation. Numerical simulations of two examples are also carried out to verify the theoretical predictions.