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

TL;DR

The paper analyzes Shilnikov's scenario in a three-dimensional setting by constructing a scaled vector field $V_\epsilon(x)=\epsilon^{-1}V(\epsilon x)$ and a two-stage return map built from inner and outer maps. Through transversality arguments and precise planar coordinates, it shows that curves expanded by the return map intersect the domain in multiple locations, enabling a symbolic dynamics description: for any binary sequence $(s_j)$ there exist trajectories with appropriate region membership and height constraints. This yields a rigorous mechanism for complicated motion near a homoclinic loop to a saddle-focus, without requiring a full horseshoe or shift conjugacy, and provides a framework for further refinement toward chaotic invariants. The approach hinges on invariant subspace structure, careful travel-time estimates, and diffeomorphic inner/outer maps that link the local and global dynamics in the plane. Overall, the work delivers a detailed, technically precise proof of Shilnikov-type complexity in 3D and lays groundwork for deeper chaotic-trajectory constructions.

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 $σ\pmμ$, $σ<0<μ$, with $0<σ+u$.

Figures (8)

• Figure 1: Actions of the return map
• Figure 2: The polar coordinates along the projected solution
• Figure 3: Parametrizing the domain of the inner map
• Figure 4: The action of the inner map and related angles for projections
• Figure 5: The inner map along vertical line segments

Theorems & Definitions (23)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 3.1
• Corollary 3.2
• Corollary 3.3
• Proposition 3.4
• Corollary 3.5
• Proposition 4.1
• Corollary 4.2