Gradient Trajectories For Plane Singular Metrics I: Oscillating Trajectories

Abstract

In this short note, we construct an example of a real plane analytic singular metric, degenerating only at the origin, such that any gradient trajectory (respectively to this singular metric) of some well chosen function spirals around the origin. The inversion mapping carries this example into an example of a gradient spiraling dynamics at infinity

Keywords

• oscillating and non-oscillating trajectories
• singular gradient
• singular metric