Embeddings of a strange attractor into ${\mathbb{R}}^3$

The algorithm for determining a global Poincaré section is applied to a previously studied dynamical system on ${\mathbb{R}}^2\times {\mathbb{S}}^1$ and a one-parameter family of embeddings of the strange attractor it generates into ${\mathbb{R}}^3$. We find that the topological properties of the attractor are embedding dependent to a limited extent. These embeddings rigidly preserve mechanism, which is a simple stretch and fold. The embeddings studied show three discrete topological degrees of freedom: parity, global torsion, and braid type of the genus-one torus bounding the embedded attractor.

Figure 1

A strange attractor generated by Eq. (1) is plotted in the $X,\dot{X}$ plane. Another attractor is obtained from a change in initial conditions: $(X,Y)\to (-X,-Y)$. Parameter values $\mu =1.0434$, $\nu =-1.0$, $ϵ=0.45$, and $\omega =0.399$.

Figure 2

Intersection of the strange attractor with a stroboscopic section (Poincaré section) at $\varphi =0.0$.

Figure 3

Return map for the stroboscopic section (Fig. 2) onto itself. The parameter $s$ is related to the arc length along the intersection of the attractor with the plane $\varphi =0$.

Figure 4

Period-one and period-two orbits in the $X\text{−}t$ plane.

Figure 5

Relative rotation rates for the period-one and period-two orbits. The difference ${\mathbf{X}}_2\left(t\right)-{\mathbf{X}}_1\left(t\right)$ is projected onto the $X\text{−}t$ plane and the number of zero crossings is counted. The relative rotation rate is half the number of zero crossings divided by the period (2).

Figure 6

Projection of the difference ${\mathbf{X}}_2\left(t\right)-{\mathbf{X}}_1\left(t\right)$ onto the $X\text{−}\dot{X}$ plane in the intrinsic solid torus ${\mathbb{R}}^2\times {\mathbb{S}}^1$. This difference rotates around the origin five times in the clockwise direction. The linking number is $-5$ and relative rotation rate is $-\frac{5}{2}$.

Figure 7

Period-one and -two orbits projected onto the $X\text{−}Y$ plane. They were obtained by finding surrogate periodic orbits, smoothing in frequency space, and first differencing in the time domain.

Figure 8

Intersections of the strange attractor generated by Eq. (1) with the plane $\dot{X}=0$ under the embedding Eq. (6). The intersections occur in the order $A1\to A2\to B1\to B2\to C1\to C2\to A1$. The flow at $A1$, $B1$, and $C1$ is into the plane while the flow at $A2$, $B2$, and $C2$ is out of the plane. (a) $\tau =225∕1024T$, (b) $\tau =425∕1024T$, (c) $\tau =-225∕1024T$.

Figure 9

When the ribbon with writhe but no twist is stretched, twist is induced. The direction of the twist depends on the coil of the ribbon.

Figure 10

Tangent vectors are drawn to the segments at each of the four crossing points, reached in the order $a$, $b$, $c$, $d$ starting from the arrow.

Figure 11

Vertical distance between the orbit segments at the four points $a$, $b$, $c$, $d$. These are points of apparent self-intersection when projected to the $X,\dot{X}$ plane. Real self-intersections occur in ${\mathbb{R}}^3$ when the vertical offset is zero.