A dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set \(\Lambda \) with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in \(\Lambda \) weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps \(\mathcal {H}\) on \(\mathbb {R}^3\), which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of intersection points of the respective one-dimensional stable manifold with a chosen planar section and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of \(\mathcal {H}\) over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of \(\mathcal {H}\).

由具有三维相空间的微分同胚给出的动力系统可能有一个混合器，它是一个双曲集\(\Lambda \)，具有一个一维稳定不变流形，其行为类似于表面。这意味着\(\Lambda \)中任何固定点或周期点的稳定流形在相空间中作为曲线来回编织，使得它在某些投影中是稠密的；我们将此称为地毯属性。我们提出了一种方法，可以高效、准确地计算这种一维流形的非常长的片段，从而可以可靠地识别大量与指定部分的交点。我们通过\(\mathbb {R}^3\)上的一系列 Hénon 类映射\(\mathcal {H}\)来证明这一点，这是唯一已知的、已证明存在的微分同胚的明确示例搅拌机的。此示例的代码可作为Matlab脚本作为补充材料。与早期的工作相比，我们的方法允许我们确定相应一维稳定流形与所选平面截面的大量交点，并在参数更改时将每条曲线渲染为单独的曲线。通过适当的精度设置，我们不仅计算相关参数区间内\(\mathcal {H}\)不动点的参数化曲线，而且还计算二周期轨道稳定流形的相应参数化曲线（具有负特征值）和三周期轨道（具有正特征值）。通过这种方式，我们证明我们的算法可以处理（最多）第四次迭代\(\mathcal {H}\)生成的大扩展率。