Embedded Duffing Oscilator

Use stereoscopic glasses for a 3D effect.

Equations: Duffing
 
$\dot{u}$ = $ v$
$\dot{v}$ = $ -\delta v -u^3+u+A\sin(\omega t)$
Parameters: $(\delta,A,\omega)=(0.4,0.4,1.0)$

The Duffing equations 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 Duffing 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 bemapped into $R^3$. One way to may the "paper towel holder" into$R^3$ is via the "natural embedding" in which the axis of the cylinder is mapped onto a circle of radius sufficient large so that the embedded strange attractor has no self intersections, for these would violate the uniqueness condition on dynamical systems. This simulation shows a mapping of the Duffing attractor into real 3-space following the trajectory of a simple circle inthe $x$-$y$ plane. The view can be considered in two ways. 1. We rotate the space containing the embedded attractor around the $x$ axis. 2. We view the fixed embedded attractor, lying mostly in the $x$-$y$ plane, from a circular orbit in the$y$-$z$ plane. Both interpretations offer the satisfying view seen here.