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

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 $x^{\prime }=V(x)$ 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 $0<\sigma +u$.

arXiv comment34 pages, 8 figures. Corrections of errors in pages/lines 3/6, 14/9, 14/bottom,17/3, 19/10, 21/12+13+14+16+18, 29/26+29+32, 30/2+5+6,31/6+11, 33/26+28+29 of the first version