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An algorithm to detect and rigorously verify blenders

Andy Hammerlindl, Natalia McAlister, Warwick Tucker

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 Hénon-like maps presents blenders.

An algorithm to detect and rigorously verify blenders

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 Hénon-like maps presents blenders.
Paper Structure (8 sections, 7 theorems, 29 equations, 8 figures, 3 tables)

This paper contains 8 sections, 7 theorems, 29 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background, definitions and proof of \ref{['theo:main']}
  3. General algorithms
  4. Characterisation of $\mathcal{U}_{\alpha} \overset{f}{\rightsquigarrow} \mathcal{V}_{\beta}$
  5. Verification for the Hénon-like system
  6. Construction of u-curves for the Hénon-like system
  7. Results
  8. Discussion

Key Result

Theorem 1.1

Let $f:\mathbb{R}^3 \to \mathbb{R}^3$ be a diffeomorphism. Let $\mathcal{D} \subset \mathbb{R}^3$ be such that $\Lambda = \bigcap_{n \in \mathbb{Z}} f^n(\mathcal{D})$ is a transitive hyperbolic set with $\dim W^u(\Lambda) = 2$. Let $\Omega$ be a non-empty family of curves that can be expressed as where $\mathcal{K}$ and $\mathcal{L}$ are finite sets of u-curves and $\mathcal{U}_{\alpha}$ and $\ma

Figures (8)

  • Figure 1: Projection onto the $xz$-plane of the objects appearing in the hypotheses of \ref{['theo:main']}, for $f$ as in \ref{['eq:H']} with $\xi = 1.2$. Top left: The u-curves $\beta \in \mathcal{L}$ (red). Top right: The u-curves $\alpha \in \mathcal{K}$ (blue). Bottom left: Images of selected subcurves of the u-curves $\alpha \in \mathcal{K}$ (blue). Bottom right: For a representative u-curve $\alpha \in \mathcal{K}$, a subset of the image of the small u-tube $\mathcal{U}_{\alpha}$, together with the big u-tube $\mathcal{V}_{\beta}$ that $\mathcal{U}_{\alpha}$ maps through.
  • Figure 2: In purple the box $\mathcal{D}$. In green its image $f(\mathcal{D})$ projected to the $xy$-plane (left) and in three dimensions (right).
  • Figure 3: The projection to the $xz$-plane of the u-tubes from \ref{['lemma:split']} for $n = 3$, $k = 2$. Above is the big u-tube around $\alpha$. Below is the same set represented as the union of the small u-tubes around $\alpha_1$ (in blue), $\alpha_2$ (in red), and $\alpha_3$ (in yellow).
  • Figure 4: Number of elements in $\mathcal{L}$ and $\mathcal{P}$ at each iteration of \ref{['alg:main']}, for $f$ as in \ref{['theo:henon']} with $\xi = 1.5$.
  • Figure 5: The projection to the $xz$-plane of a through box $B$ for the u-curves $\alpha$ and $\beta$ (\ref{['deff:through_box']}).
  • ...and 3 more figures

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Hyperbolic set
  • Definition 2.2: Stable and unstable manifolds
  • Definition 2.3: cu-blender
  • Proposition 2.4
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof : Proof of \ref{['prop:fam']}
  • ...and 12 more