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

Shilnikov’s scenario in 3D consists of a vectorfield $V$ so that the equation $$x^{\mathrm{\prime }}(t)=V(x(t))\in {\mathbb{R}}^3$$ 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 .