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

Hans-Otto Walther

TL;DR

This work extends Shilnikov's chaotic scenario to 3D vector fields of minimal smoothness ($C^1$) by developing a flow-based framework around a homoclinic loop. It introduces scaled flows $F_{\epsilon}$ and decomposes the dynamics into inner and exterior maps, using angle tracking in the stable plane to formulate a planar return map whose geometry supports symbolic dynamics. The authors prove the existence of topological chaos by showing that the planar return map encodes arbitrary binary sequences, yielding a countable family of chaotic trajectories and extending prior results to $C^1$ vector fields. The approach relies on a careful flow transformation (Appendix) that preserves smoothness and enables precise transversality, exponential estimates, and constructive estimates for the return map, while outlining open problems such as periodic orbits and extensions to higher dimensions.

Abstract

Shilnikov's scenario in 3D consists of a vectorfield $V$ so that the equation $$ x'(t)=V(x(t))\in\mathbb{R}^3 $$ with $V(0)=0$ has a solution homoclinic to the origin and the eigenvalues of $DV(0)$ are $u>0$ and $σ\pm iμ$, $σ<0<μ$, with $0<σ+u$. 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 .

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

TL;DR

This work extends Shilnikov's chaotic scenario to 3D vector fields of minimal smoothness () by developing a flow-based framework around a homoclinic loop. It introduces scaled flows and decomposes the dynamics into inner and exterior maps, using angle tracking in the stable plane to formulate a planar return map whose geometry supports symbolic dynamics. The authors prove the existence of topological chaos by showing that the planar return map encodes arbitrary binary sequences, yielding a countable family of chaotic trajectories and extending prior results to vector fields. The approach relies on a careful flow transformation (Appendix) that preserves smoothness and enables precise transversality, exponential estimates, and constructive estimates for the return map, while outlining open problems such as periodic orbits and extensions to higher dimensions.

Abstract

Shilnikov's scenario in 3D consists of a vectorfield so that the equation with has a solution homoclinic to the origin and the eigenvalues of are and , , with . We give a detailed proof that close to the homoclinic loop complicated motion exists provided 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 .
Paper Structure (9 sections, 26 theorems, 196 equations, 6 figures)

This paper contains 9 sections, 26 theorems, 196 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Transformation, scaling, projected flowlines
  3. Angles
  4. Transversality, and exterior maps
  5. Inner maps
  6. The return map in the plane
  7. Curves expanded by the return map in the plane
  8. Complicated dynamics
  9. Appendix: From the vectorfield $V$ to the flow $F$

Key Result

Proposition 2.1

(i) For every $\eta>0$ there exists $\epsilon(\eta)>0$ such that for all $\epsilon\in(0,\epsilon(\eta))$, (ii) There exists $\epsilon_B>0$ such that $0<\epsilon<\epsilon_B$, and $L$ and $U$ are invariant under $F_{\epsilon}$ in $B$.

Figures (6)

  • Figure 1: Top: The return map as a composition of the inner map with the exterior map. Bottom: Angles from Proposition 3.1
  • Figure 2: Top: The relations (2). Bottom: The exterior map.
  • Figure 3: Top: The inner map and related angles. Bottom: The inner map along vertical line segments.
  • Figure 4: The map $\Phi_j\circ g$
  • Figure 5: Positions of $Q_j(\psi,\delta)$ depending on the angle $\Phi_j(\psi,\delta)$
  • ...and 1 more figures

Theorems & Definitions (26)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Proposition 4.1
  • Proposition 4.2
  • Corollary 4.3
  • ...and 16 more