Everything about Henon Map totally explained
The
Hénon map is a discrete-time
dynamical system. It is one of the most studied examples of dynamical systems that exhibit
chaotic behavior. The Hénon map takes a point (
x,
y) in the plane and maps it to a new point
» .
The map depends on two parameters,
a and
b, which for the
canonical Hénon map have values of
a = 1.4 and
b = 0.3. For the canonical values the Hénon map is chaotic. For other values of
a and
b the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the type of behavior of the map at different parameter values may be obtained from its
orbit diagram.
The map was introduced by
Michel Hénon as a simplified model of the
Poincaré section of the
Lorenz model. For the canonical map, an initial point of the plane will either approach a set of points known as the Hénon
strange attractor, or diverge to infinity. The Hénon attractor is a
fractal, smooth in one direction and a
Cantor set in another. Numerical estimates yield a
correlation dimension of 1.42 ± 0.02 and a
Hausdorff dimension of 1.261 ± 0.003 for the attractor of the canonical map.
As a dynamical system, the canonical Hénon map is interesting because, unlike the
logistic map, its orbits defy a simple description.
Attractor
The Hénon maps two points into themselves: these are the invariant points. For the canonical values of
a and
b of the Hénon map, one of these points is on the attractor:
» x = 0.631354477... and
y = 0.189406343...
This point is unstable. Points close to this fixed point and along the slope 1.924 will approach the fixed point and points along the slope -0.156 will move away from the fixed point. These slopes arise from the linearizations of the
stable manifold and
unstable manifold of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the
strange attractor of the Hénon map.
The Hénon map doesn't have a strange attractor for all values of the parameters
a and
b. For example, by keeping
b fixed at 0.3 the bifurcation diagram shows that for
a = 1.25 the Hénon map has a stable periodic orbit as an attractor.
Cvitanović et al. have shown how the structure of the Hénon strange attractor can be understood in terms of unstable periodic orbits within the attractor.
Decomposition
The Hénon map may be decomposed into an area-preserving bend:
» ,
a contraction in the
x direction:
» ,
and a reflection in the line
y =
x:
» .
Further Information
Get more info on 'Henon Map'.
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