Jordan curves and funnel sections

TL;DR

This paper explores the properties of solution funnels in non-unique ODE solutions, constructs a metric space of Jordan curves, and shows that most Jordan curves cannot be pierced by finite-length arcs.

Contribution

It introduces a natural complete metric on Jordan curves and analyzes the generic properties of these curves in the context of solution funnels for differential equations.

Findings

The space of Jordan curves can be equipped with a natural complete metric.

Most Jordan curves are nowhere pierceable by finite-length arcs.

The study provides insights into the structure of solution sets for non-unique ODEs.

Abstract

We study the case when solution of an ODE at a given initial condition fail to be unique and investigate what are the possible time-1 sections of the `solution funnel'. Along the way we give construction of a natural complete metric on the space of Jordan curves and prove that the generic Jordan curve is nowhere pierceable by arcs of finite length.