July 21, 2013

For the parameters usually studied, the Lorenz system,

has a single global strange attractor. However, there is a small region of parameter space,

24.06 <

where the strange attractor coexists with two stable equilibria. This is mentioned in Strogatz,

Since the basins are three-dimensional, they are shown here in two-dimensional cross sections. The first plot shows a cross section in the plane

Basins of attraction for the Lorenz system
with *r* = 24.4 in the plane *z* = 23.4

Since the above cross section does not contain the unstable equilibrium at the origin, the following plot show the basins in the*x=y* plane where all three equilibria
lie, with the unstable one indicated by an open circle. Any
resemblance to a smiley face is coincidental.

Basins of attraction for the Lorenz system with*r*
= 24.4 in the plane *x=y*

Since the above cross section does not contain the unstable equilibrium at the origin, the following plot show the basins in the

Basins of attraction for the Lorenz system with

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