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

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 $x^{\mathrm{\prime }}=V(x)$ has a homoclinic solution $h$ with ${\lim }_{\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }}h(t)=0$, and $DV(0)$ has eigenvalues $u>0$ and $\sigma \pm \mu$, $\sigma <0<\mu$, with $0<\sigma +u$.