Differential positivity characterizes one-dimensional normally hyperbolic attractors

TL;DR

This paper demonstrates that one-dimensional normally hyperbolic attractors in smooth dynamical systems can be characterized by differential positivity, linking geometric properties to contraction of cone fields.

Contribution

It introduces a novel characterization of one-dimensional normally hyperbolic attractors via differential positivity, extending the understanding of hyperbolic attractors.

Findings

Normal hyperbolic attractors are characterized by differential positivity.

Differential positivity involves contraction of a cone field.

Analogy with zero-dimensional hyperbolic attractors and differential stability.

Abstract

The paper shows that normally hyperbolic one-dimensional compact attractors of smooth dynamical systems are characterized by differential positivity, that is, the pointwise infinitesimal contraction of a smooth cone field. The result is analog to the characterization of zero-dimensional hyperbolic attractors by differential stability, which is the pointwise infinitesimal contraction of a Riemannian metric.