Article Contents
Article Contents

# How to identify a hyperbolic set as a blender

• A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

Mathematics Subject Classification: Primary: 37D25, 37D10; Secondary: 37C05, 37G25, 37M21.

 Citation:

• Figure 1.  Illustration of the hyperbolic set $\Lambda_h$ (black dots) as the closure of the intersection between the manifolds $W^s(p_h^\pm)$ (blue curves) and $W^u(p_h^\pm)$ (red curves) of the saddle fixed points $p_h^\pm$ (green crosses); panels (a) and (b) show two views of the $(x, y)$-plane, and panel (c) shows the Poincaré disk in the $(\bar{x}, \bar{y})$-plane

Figure 2.  The hyperbolic set $\Lambda$ (black dots) of $H$ with $\xi = 1.2$, determined as the intersection set of $W^s(p^-)$ (dark blue) and $W^s(p^+)$ (light blue) with $W^u(p^-)$ (red surface), shown in $(\bar{x}, \bar{y}, \bar{z})$-space (a) and in projection onto the $(\bar{x}, \bar{z})$-plane (b). Panels (c) and (d) illustrate $\Lambda$ and its tangent space $T^s(\Lambda)$ (green lines) in $(x, y, z)$-space and in projection onto the $(x, z)$-plane, respectively; four regions are highlighted with different shades of green

Figure 3.  The hyperbolic set $\Lambda$ (black dots) of $H$ with $\xi = 2.0$, determined as the intersection set of $W^s(p^-)$ (dark blue) and $W^s(p^+)$ (light blue) with $W^u(p^-)$ (red surface), shown in $(\bar{x}, \bar{y}, \bar{z})$-space (a) and in projection onto the $(\bar{x}, \bar{z})$-plane (b). Panels (c) and (d) illustrate $\Lambda$ and its tangent space $T^s(\Lambda)$ (green lines) in $(x, y, z)$-space and in projection onto the $(x, z)$-plane, respectively; four different regions are highlighted with different shades of green

Figure 4.  The hyperbolic set $\Lambda$ of $H$ with $\xi = 0.8$, determined as the intersection set of $W^u(p^-)$ (red curves) and $W^u(p^+)$ (magenta curves) with $W^s(p^-)$ (blue surface), shown in $(\bar{x}, \bar{y}, \bar{z})$-space (a) and in projection onto the $(\bar{y}, \bar{z})$-plane (b). Panels (c) and (d) illustrate $\Lambda$ and its tangent space $T^u(\Lambda)$ (green lines) in $(x, y, z)$-space and in projection onto the $(y, z)$-plane, respectively; four regions are highlighted with different shades of green

Figure 5.  The hyperbolic set $\Lambda$ of $H$ with $\xi = 0.45$, determined as the intersection set of $W^u(p^-)$ (red curves) and $W^u(p^+)$ (magenta curves) with $W^s(p^-)$ (blue surface), shown in $(\bar{x}, \bar{y}, \bar{z})$-space (a) and in projection onto the $(\bar{y}, \bar{z})$-plane (b). Panels (c) and (d) illustrate $\Lambda$ and its tangent space $T^u(\Lambda)$ (green lines) in $(x, y, z)$-space and in projection onto the $(y, z)$-plane, respectively; four regions are highlighted with different shades of green

Figure 6.  The intersection set for $\xi = 1.2$ of the stable manifolds $W^s(p^-)$ (dark blue) and $W^s(p^+)$ (light blue) with the section $\Sigma$ (gray plane) defined by $\bar{x} = \bar{y}$. Panel (a) shows the intersection points in $\Sigma$ and panel (b) shows how $W^s(p^\pm)$ intersect $\Sigma$ in $(\bar{x}, \bar{y}, \bar{z})$-space

Figure 7.  The intersection set for $\xi = 2.0$ of the stable manifold $W^s(p^-)$ (dark blue) and $W^s(p^+)$ (light blue) with the section $\Sigma$ (gray plane) defined by $\bar{x} = \bar{y}$. Panel (a) shows the intersection points in $\Sigma$ and panel (b) shows how $W^s(p^\pm)$ intersect $\Sigma$ in $(\bar{x}, \bar{y}, \bar{z})$-space

Figure 8.  The five largest $\bar{z}$-gaps $\Delta^i$, for $i = 1, \dots, 5$, of $W^s(p^-)$ in $\Sigma$ as a function of the arclength, represented by the exponent $k$, for $\xi = 1.2$ (a1) and for $\xi = 2.0$ (a2). Panel (b) shows the largest gap $\Delta^1$ versus $\xi$ for $k = 6$ (red) when $0 < \xi <1$ and for $k = 7$ (blue) when $1 < \xi$. Panel (c) shows the associated $\bar{z}$-values of $p^\pm$ (green), $W^s(p^-) \cap \Sigma$ (blue) and $W^u(p^-) \cap \Sigma$ (red), respectively

Figure 9.  Self-similar structure of the intersection set $W^s(p^-) \cap \Sigma$ for $\xi = 1.2$. Panel (a1) shows a part of $W^s(p^-) \cap \Sigma$ in a color coding according to the $\bar{x}$-values, and panel (a2) is an enlargement. Panels (b1) and (b2) show $\bar{x}_{n+1}$ versus $\bar{x}_{n}$ and $\bar{z}_{n+1}$ versus $\bar{z}_{n}$, respectively, of successive points of $W^s(p^-) \cap \Sigma$

Figure 10.  Self-similar structure of the intersection set $W^s(p^-) \cap \Sigma$ for $\xi = 2.0$. Panel (a1) shows a part of $W^s(p^-) \cap \Sigma$ in a color coding according to the $\bar{x}$-values, and panel (a2) is an enlargement. Panels (b1) and (b2) show $\bar{x}_{n+1}$ versus $\bar{x}_{n}$ and $\bar{z}_{n+1}$ versus $\bar{z}_{n}$, respectively, of successive points of $W^s(p^-) \cap \Sigma$

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