# A geometric criterion for the existence of chaos based on periodic orbits in continuous-time autonomous systems

## TL;DR

This paper introduces a novel geometric criterion for chaos in 3D continuous systems, focusing on the intersection properties of stable and unstable manifolds of hyperbolic periodic orbits, expanding beyond classical criteria.

## Contribution

It presents a new geometric approach to determine chaos based on manifold intersections, distinct from traditional equilibrium or homoclinic-based criteria.

## Key findings

- Establishes a criterion involving hyperbolic periodic orbits and manifold intersections.
- Shows the existence of a Smale horseshoe under specific geometric conditions.
- Provides a framework that extends classical chaos criteria beyond equilibrium points.

## Abstract

A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on cross-sections of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic or heteroclinic orbits of Poincar\'{e} maps.