The cubic Henon map
is a discrete dynamical system in the plane. The map is parameterized by a constant g.
Below you see some orbits of the map for two different values of g.
The point (0,0) is always an equilibrium point. As in the case of linear
dynamical systems, one is interested about the stability of this fixed point. One can ask for example, which
points stay for every in a bounded region and do not escap to infinity. In the picture to
the left, you clearly see a large island around (0,0) which seems to stay, while in the
picture to the right, a point starting near (0,0) escapes to infinity in general.
Evenso this stability question is difficult, linear algebra helps to understand the
system near the equilibrium point. For small x, the linear approximation of the map
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T(x,y) = ( g x-y,x) = A(x,y)
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gives trajectories close to the real trajectories. We can analyse this linear system
as we have learned: because the determinant D is 1 and the trace is g, the eigenvalues
are g/2-(g2/4-1)1/2 and g/2-(g2/4-1)1/2.
For g>2 there are two real eigenvalues, one smaller than 1, one bigger than 1 and the
trajectories ly on hyperbola, and orbits escape to infinity. One can show that the
nonlinear system also has curves invariant which ressemble the hyperbola in the linear case.
For |g| smaller than 2, there are two complex eigenvalues of length 1. The linear system has
then trajectories, which ly on ellipses. One can show using a theory called KAM that also the
nonlinear system has curves invariant near the equilibrium point.
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