The Hénon map is a discrete-time dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

. The Hénon map takes a point (x_n, y_n) 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

Michel Hénon

Michel Hénon is a mathematician and astronomer. He is currently at the Nice Observatory.In astronomy, Hénon is well known for his contributions to stellar dynamics. In late 1960s and early 1970s he was involved in dynamical evolution of star clusters, in particular the globular clusters...

  as a simplified model of the Poincaré section

Poincaré map

In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincaré section, transversal to...

 of the Lorenz model

Lorenz attractor

The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...

. 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

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

, smooth in one direction and a Cantor set

Cantor set

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....

 in another. Numerical estimates yield a correlation dimension

Correlation dimension

In chaos theory, the correlation dimension is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension....

 of 1.25 ± 0.02 and a Hausdorff dimension

Hausdorff dimension

thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...

 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

Logistic map

The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...

, its orbits defy a simple description.

Attractor

The Hénon map 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:

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

Stable manifold

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor...

 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 does not 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:

.

See also

• Fractal
  Fractal
  A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
  
  
• List of chaotic maps
• Smale's horseshoe map
  Horseshoe map
  In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator...
  
  
• Takens' theorem
  Takens' theorem
  In mathematics, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system...
  
  

External links

• Interactive Henon map and Henon attractor in Chaotic Maps
• Another interactive iteration of the Henon Map by A. Luhn
• Orbit Diagram of the Hénon Map by C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr, The Wolfram Demonstrations Project.