Hénon map

thumb|Hénon attractor for and

In mathematics, 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 in the plane and maps it to a new point:<math display="block">\begin{cases}x_{n+1} = 1 - a x_n^2 + y_n\\y_{n+1} = b x_n\end{cases}</math>The map depends on two parameters, and , which for the classical Hénon map have values of and . For the classical values, the Hénon map is chaotic. For other values of and , the map may be chaotic, intermittent, or converge to a periodic orbit. An overview of the map's behavior at different parameter values can be seen in its orbit diagram.

The map was introduced by Michel Hénon as a simplified model for the Poincaré section of the Lorenz system. and a box-counting dimension of 1.261 ± 0.003.

Dynamics

The Attractor

The Hénon map is a two-dimensional diffeomorphism with a constant Jacobian determinant. The Jacobian matrix of the map is:<math display="block">J = \begin{bmatrix} -2ax & 1 \\ b & 0 \end{bmatrix}</math>The determinant of this matrix is <math>det(J) = -b</math>. Because the map is dissipative (i.e., volumes shrink under iteration), the determinant must be between -1 and 1. The Hénon map is dissipative for . For the classical parameters <math>a = 1.4, b = 0.3</math>, the determinant is -0.3, so the map contracts areas at a constant rate. Every iteration shrinks areas by a factor of 0.3.

This contraction, combined with a stretching and folding action, creates the characteristic fractal structure of the Hénon attractor. For the classical parameters, most initial conditions lead to trajectories that outline this boomerang-like shape. The attractor contains an infinite number of unstable periodic orbits, which are fundamental to its structure.

Fixed points

The map has two fixed points, which remain unchanged by the mapping. These are found by solving and . Substituting the second equation into the first gives the quadratic equation:<math display="block">ax^2 + (1-b)x - 1 = 0</math>The solutions (the x-coordinates of the fixed points) are:<math display="block">x = \frac{-(1-b) \pm \sqrt{(1-b)^2 + 4a{2a}</math>For the classical parameters and , the two fixed points are:

<math>x_1 \approx 0.631, \quad y_1 \approx 0.189</math>

<math>x_2 \approx -1.131, \quad y_2 \approx -0.339</math>

The stability of these points is determined by the eigenvalues of the Jacobian matrix evaluated at the fixed points. For the classical map, the first fixed point is a saddle point (unstable), while the second fixed point is a repeller (also unstable). The unstable manifold of the first fixed point is a key component that generates the strange attractor itself.

For chaotic systems like the Hénon map, the eigenfunctions are typically complex, fractal-like functions. They cannot be found analytically and must be computed numerically, often using methods like Dynamic Mode Decomposition (DMD). The level sets of the Koopman modes can reveal the invariant structures of the system, such as the stable and unstable manifolds and the basin of attraction, providing a global picture of the dynamics.

Decomposition

thumb|right|The classical Hénon map after 15 iterations, showing the stretching and folding action described by the decomposition.

The Hénon map can be decomposed into a sequence of three simpler geometric transformations. This helps to understand how the map stretches, squeezes, and folds phase space. Their physical, experimental approach to the Lorenz system led to two key insights. First, they identified a transition where the system switches from a strange attractor to a limit cycle at a critical parameter value. This phenomenon would later be explained by Pomeau and Paul Manneville as the "scenario" of intermittency.

Second, Pomeau and Ibanez suggested that the complex dynamics of the three-dimensional, continuous Lorenz system could be understood by studying a much simpler, two-dimensional discrete map that possessed similar characteristics.

Generalizations

3D Hénon map

A 3-D generalization for the Hénon map was proposed by Hitzl and Zele:

:<math>\mathbf{s}(n+1)=

\begin{bmatrix}

s_1(n+1) \\

s_2(n+1) \\

s_3(n+1)

\end{bmatrix}

\begin{bmatrix}

1 - \alpha s_1^2(n) + s_3(n) \\

-\beta s_1(n) \\

\beta s_1(n) + s_2(n)

\end{bmatrix}</math>

For certain parameters (e.g., <math>\alpha=1.07</math> and <math>\beta=0.3</math>), this map generates a chaotic attractor.

References

External links

Interactive Hénon map from ibiblio.org
Orbit Diagram of the Hénon Map from The Wolfram Demonstrations Project.
Simulation of the Hénon map in javascript from CNRS.