or

The Lambda Lemma in Action

This illustrates a *generic unfolding of a homoclinic tangency*:
Specifically, a theorem from dynamical systems says that (generically)
every homoclinic tangency is an accumulation point of homoclinic
tangencies in parameter space. You can watch the homoclinic tangencies
form from the unstable manifold, because it loops around close to
itself again and again (by the lambda lemma).

The movie repeats forever.

Further: Below is picture of the whole region x=0..1, y=0.5..1.5 for parameter k=0.84, followed by a few iterates for this parameter to show you the dynamics of the map.

**Credits:**

Starring: The Standard Map

Director: Evelyn Sander

Camera operator: DSTool

Editing and Production: GifMerge

Theorem: Palis and Takens, *Hyperbolicity and sensitive
chaotic dynamics at homoclinic bifurcations*.

Filmed on location on the Surface of the Cylinder

Caterer for Dr. Sander: Thomas Wanner

Parameters b=1, k=0.82..0.84

Approximate Zoom x=0.5..0.65, y=0.85..1

The Equation

x1 = x + y1

y1 = b*y - k/(2*pi)*(sin(2*pi*x))