An example of a vector field with the oriented shadowing property

TL;DR

This paper presents an example of a vector field that exhibits the oriented shadowing property but lacks the standard shadowing property, highlighting differences in shadowing behaviors under various reparametrizations.

Contribution

The paper provides the first known example of a vector field with the oriented shadowing property without possessing the standard shadowing property.

Findings

Existence of a vector field with oriented shadowing but not standard shadowing

Differentiation between shadowing properties under reparametrizations

Illustration of shadowing property distinctions in vector fields

Abstract

We consider shadowing properties for vector fields corresponding to different type of reparametrisations. We give an example of a vector field which has the oriented shadowing properties, but does not have the standard shadowing property.