Van Der Pol

Equations: van der Pol (Shaw version)
 
$\dot{u}$ = $ bv+(c-dv^2)u$
$\dot{v}$ = $ -u+A\sin(\omega t)$
Parameters: $(b,c,d,A,\omega)=(0.7,1.0,10.0,0.25,\pi/2)$

The van der Pol equations (as modified by Shaw) are integrated for $N$ periods. In the limit $N \rightarrow \infty$ the "$\Omega$ limit set" that is created is a periodic strange attractor (strange as that may sound, it really is!). This strange attractor lives in a torus, $D^2 \times S^1$. Basically, this can be visualized as the cylinder around which paper towels are wrapped, and the output at one end is identical to the input at the other. This is a consequence of periodic boundary conditions.

The ambient space containing the van der Pol attractor (cylinder with periodic boundary conditions) can be mapped into an ambient space with a different global topology in many ways: as many ways as the cylinder $D^2 \times S^1$ can be mapped into $R^3$. One way to map the "paper towel holder" into $R^3$ is via the "natural embedding" in which the axis ofthe cylinder is mapped onto a circle of radius sufficiently large so that the embedded strange attractor has no self intersections, for these would violate the uniqueness condition on dynamical systems.

In the simulation at the bottom center the cylinder (with periodic boundary conditions) is mapped into $R^3$ using the "natural embedding". This means that the cylinder is mapped into $R^3$ in such a way that the axis of the cylinder lies along a circle of suitable radius in the $x$-$y$ plane. When a surface of section is swept around the $z$-axis the intersection appears as shown. This is exactly the same as the intersection in the ambient space $D^2 \times S^1$ when the plane defined by $\phi \in S^1$is swept from $0$ to $2\pi$ radians, or $t$ (time) swept from $0$ to $T=2\pi/\omega$.

If the cylinder is rotated by $2\pi$ radians as it is mapped into $R^3$ the two-fold symmetry of the van der Pol attractor under $(u,v,t) \rightarrow (-u,-v,t + \frac{1}{2}T)$ is removed. These two natural embeddings with rotation by $\pm 2\pi$ radians are shown in the simulations at the bottom to the left and right of the natural embedding without rotation. These are projections into counter-rotating van der Pol planes: the intersections themselves appear to counter-rotate.

In the simulation at the top the embedding of the cylinder $D^2 \times S^1$ into $R^3$ follows the guiding curve of a trefoil knot. The knot goes around the $z$-axis 3 times before closing up. Shown is the intersection of the embedded strange attractor with a plane hinged on the $z$-axis as the plane rotates around that axis.