On Shilnikov's scenario with a homoclinic orbit in 3D

TL;DR

This paper rigorously proves the existence of complex dynamics in a 3D vector field scenario involving a homoclinic orbit, extending Shilnikov's theory with detailed mathematical validation.

Contribution

It provides a detailed proof confirming the existence of complicated motions in a 3D vector field with a homoclinic orbit, enriching the understanding of Shilnikov's scenario.

Findings

Existence of homoclinic solutions in 3D vector fields.

Validation of complex dynamics near homoclinic orbits.

Mathematical conditions for Shilnikov's scenario in smooth systems.

Abstract

The paper provides a detailed proof that complicated motion exists in Shilnikov's scenario of a smooth vectorfield $V$ on $mathbb{R}^3$ with $V(0)=0$ so that the equation $x'=V(x)$ has a homoclinic solution $h$ with $\lim_{|t|\to\infty}h(t)=0$, and $DV(0)$ has eigenvalues $u>0$ and $\sigma\pm\mu$, $\sigma<0<\mu$, with $0<\sigma+u$.