Topological aspects of the structure of chaotic attractors in ${\mathbb{R}}^3$

Strange attractors with Lyapunov dimension $d_L<3$ can be classified by branched manifolds. They can also be classified by the bounding tori that enclose them. Bounding tori organize branched manifolds (classes of strange attractors) in the same way as the branched manifolds organize the periodic orbits in a strange attractor. We describe how bounding tori are constructed and expressed in a useful canonical form. We present the properties of these canonical forms and show that they can be uniquely coded by analogs of periodic orbits of period $g-1$, where $g$ is the genus. We describe the structure of the global Poincaré surface of section for an attractor enclosed by a genus-$g$ torus and determine the transition matrix for flows between the $g-1$ components of the Poincaré surface of section. Finally, we show how information about a bounding torus can be extracted from scalar time series.

Figure 1

Solution of the Lorenz system of ODEs approaching a strange attractor. Control parameter values: $\left(\sigma ,r,b\right)=\left(10,28,8∕3\right)$.

Figure 2

Transformation of the branched manifold for the Lorenz system into a different form by applying a series of local moves.

Figure 3

The branched manifold at the bottom of Fig. 2 is shown enclosed by a bounding torus of genus 3. All three fixed points in the vector field generating the flow are outside the bounding torus. The flow around the two foci, as well as the regular saddle that separates them, is shown. The four singularities on the surface induced by the flow in the neighborhood of the central hole are shown as dots and the flow directions in the neighborhood of these four singularities are shown. All four singularities are regular saddles.

Figure 4

(a) Two-dimensional genus three surface $\Omega$ that has an orientation induced by the flow around the focal points. (b) Planar equivalent of $\Omega$ indicating the singular set of the induced flow. (c) Canonical form.

Figure 5

(a) Branched manifold describing the three-fold cover of the Rössler system. (b) Branched manifold for attractor with no symmetry. (c) Canonical form for both systems.

Figure 6

Holes with (a) 0, (b) 2, (c) 4, and (d) 6 singularities. Only holes with two singularities can be removed without changing the asymptotic structure of the flow.

Figure 7

Canonical forms corresponding to genus-8 bounding torus. They all have five circles, two squares, and one hexagon. The three forms are inequivalent—they are described by different periodic sequences: (a) abbacca, (b) abccbaa, and (c) abbccaa.

Figure 8

Longitudes $L_i$ and meridians $M_i$ for a handle-body of genus $g$.

Figure 9

Chaotic data set.

Figure 10

Differential embedding of the data set shown in Fig. 8. The axes are $\left(x\left(t\right),\dot{x}\left(t\right)\right)$. The fixed points are along the abscissa. The locations of suitably chosen components of the Poincaré surface of section are shown and numbered from 1 to 3, the first being the leftmost one.

Figure 11

A projection on the $XY$ plane of the strange attractor of the threefold cover of the Rössler system.