Gradient trajectories for plane singular metrics I: oscillating trajectories

TL;DR

This paper constructs a specific example of a real plane singular metric where gradient trajectories spiral around the origin, illustrating complex dynamical behavior near singularities and at infinity.

Contribution

It introduces a novel example of a singular metric with spiraling gradient trajectories, highlighting intricate geometric and dynamical properties.

Findings

Gradient trajectories spiral around the origin.

Inversion mapping transforms the behavior to infinity.

Demonstrates complex dynamics near singularities.

Abstract

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.