An algorithm to detect and rigorously verify blenders

TL;DR

This paper introduces an algorithmic framework for verifying blenders in dynamical systems using computer-assisted methods, applicable even with rough approximations of unstable directions.

Contribution

The authors develop a systematic, computer-verified method to identify blenders in dynamical systems without requiring precise invariant manifold data.

Findings

Successfully verified blenders in a family of three-dimensional Hénon-like maps.

The method requires only rough approximations of unstable directions, simplifying verification.

Provides a general framework for computer-assisted verification of complex dynamical structures.

Abstract

We present a characterisation of blenders based on mapping properties of certain sets of curves that can be rigorously verified by computer-assisted methods. We develop an algorithm to construct these sets of curves that requires only a rough approximation of the strong unstable direction in a prescribed region. Since our approach does not rely on precise data, such as the exact location of invariant manifolds or fixed points, it provides a systematic framework to verify blenders in explicit examples. Here, we apply this framework to rigorously verify that a family of three-dimensional H\'enon-like maps presents blenders.