On Shilnikov's scenario in 3D: Topological chaos for vectorfields of class $C^1$

TL;DR

This paper proves that in 3D vector fields with a homoclinic loop, complex chaotic motion can occur under minimal smoothness assumptions, specifically for once continuously differentiable vector fields, extending previous results.

Contribution

It provides a detailed proof of topological chaos near homoclinic loops in 3D vector fields with only $C^1$ smoothness, using flow-based methods instead of ODE techniques.

Findings

Existence of complex motion near homoclinic loops in $C^1$ vector fields

Extension of Shilnikov's scenario to less smooth vector fields

Major methodological modifications for flow-based analysis

Abstract

Shilnikov's scenario in 3D consists of a vectorfield $V$ so that the equation $$ x'(t)=V(x(t))\in\mathbb{R}^3 $$ with $V(0)=0$ has a solution homoclinic to the origin and the eigenvalues of $DV(0)$ are $u>0$ and $\sigma\pm i\mu$, $\sigma<0<\mu$, with $0<\sigma+u$. We give a detailed proof that close to the homoclinic loop complicated motion exists provided $V$ is just once continuously differentiable. The result requires working with flows instead of an ODE, which necessitates major modifications compared to the earlier approach for twice continuously differentiable vectorfields in arXiv:2406.18289 .