Hénon map
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 Template:Math 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, Template:Mvar and Template:Mvar, which for the classical Hénon map have values of Template:Math and Template:Math.<ref name="Henon1976">Template:Cite journal</ref> For the classical values, the Hénon map is chaotic. For other values of Template:Mvar and Template:Mvar, 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.<ref name="Henon1976" /> For the classical map, an initial point in the plane will either approach a set of points known as the Hénon strange attractor, or it will diverge to infinity. The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another.<ref name="Henon1976" /> Numerical estimates for the fractal dimension of the strange attractor for the classical map yield a correlation dimension of 1.21 ± 0.01<ref name="Grassberger1983">Template:Cite journal</ref> and a box-counting dimension of 1.261 ± 0.003.<ref name="Russell1980">Template:Cite journal</ref>
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 Template:Math.<ref name="Alligood1996">Template:Cite book</ref> 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.<ref name="Cvitanovic1988">Template:Cite journal</ref>
Fixed points
The map has two fixed points, which remain unchanged by the mapping. These are found by solving Template:Math and Template:Math. 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 Template:Math and Template:Math, 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 Template:Mvar 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).<ref name="Strogatz2015">Template:Cite book</ref> The unstable manifold of the first fixed point is a key component that generates the strange attractor itself.<ref name="Strogatz2015" />
Bifurcation diagram
The Hénon map exhibits complex behavior as its parameters are varied. A common way to visualize this is with a bifurcation diagram. If Template:Mvar is held constant (e.g., at 0.3) and Template:Mvar is varied, the map transitions from regular (periodic) to chaotic behavior. This transition occurs through a period-doubling cascade, similar to that of the logistic map.<ref name="Alligood1996" />
For small values of Template:Mvar, the system converges to a single stable fixed point. As Template:Mvar increases, this point becomes unstable and splits into a stable 2-cycle. This cycle then becomes unstable and splits into a 4-cycle, then an 8-cycle, and so on, until a critical value of Template:Mvar is reached where the system becomes fully chaotic. Within the chaotic region, there are also "windows" of periodicity where stable orbits reappear for certain ranges of Template:Mvar.<ref name="Strogatz2015" />
Koopman operator analysis
An alternative way to analyze dynamical systems like the Hénon map is through the Koopman operator method. This approach offers a linear perspective on nonlinear dynamics. Instead of studying the evolution of individual points in phase space, one considers the action of the system on a space of "observable" functions, Template:Math. The Koopman operator, Template:Mvar, is a linear operator that maps an observable Template:Mvar to its value at the next time step:<math display="block">(Ug)(\mathbf{x}_n) = g(\mathbf{x}_{n+1}) = g(1-ax_n^2+y_n, bx_n)</math>While the operator Template:Mvar is linear, it acts on an infinite-dimensional function space. The key to the analysis is to find the eigenfunctions Template:Math and eigenvalues Template:Math of this operator, which satisfy Template:Math. These eigenfunctions, also known as Koopman modes, and their corresponding eigenvalues contain significant information about the system's dynamics.<ref>Template:Cite journal</ref>
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).<ref>Template:Cite journal</ref> 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.<ref>Template:Cite journal</ref>
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.<ref name="Henon1976" /> The map Template:Math can be seen as the composition Template:Math of three functions:
- Bending: An area-preserving nonlinear bend in the Template:Mvar direction:
- <math>(x_1, y_1) = (x, 1 - ax^2 + y)</math>
- Contraction: A contraction in the Template:Mvar direction:
- <math>(x_2, y_2) = (bx_1, y_1)</math>
- Reflection: A reflection across the line Template:Math:
- <math>(x_3, y_3) = (y_2, x_2)</math>
The final point is Template:Math. This decomposition separates the area-preserving folding action (step 1) from the dissipative contraction (step 2).<ref name="Alligood1996" />
History
In 1976, the physicist Yves Pomeau and his collaborator Jean-Luc Ibanez undertook a numerical study of the Lorenz system. By analyzing the system using Poincaré sections, they observed the characteristic stretching and folding of the attractor, which was a hallmark of the work on strange attractors by David Ruelle.<ref>Template:Cite book</ref> 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.<ref>Template:Cite journal</ref>
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.<ref name="Henon1976"/> In January 1976, Pomeau presented this idea at a seminar at the Côte d'Azur Observatory. Michel Hénon, an astronomer at the observatory, was in attendance. Intrigued by the suggestion, Hénon began a systematic search for the simplest possible map that would exhibit a strange attractor. He arrived at the now-famous quadratic map, publishing his findings in the seminal paper, "A two-dimensional mapping with a strange attractor."<ref name="Henon1976"/><ref>Template:Cite web</ref>
Generalizations
3D Hénon map
A 3-D generalization for the Hénon map was proposed by Hitzl and Zele:<ref name="Hitzl1985">Template:Cite journal</ref>
- <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.<ref name="Hitzl1985" />
Four-dimensional extension
The Hénon map can be plotted in four-dimensional space by treating its parameters, a and b, as additional axes. This allows for a visualization of the map's behavior across the entire parameter space. One way to visualize this 4D structure is to render a series of 3D slices, where each slice represents a fixed value of one parameter (e.g., a) while the other three (x, y, b) are displayed. The fourth parameter is then varied as a time variable, creating a video of the evolving 3D structure.
Filtered Hénon map
Other generalizations involve introducing feedback loops with digital filters to create complex, band-limited chaotic signals.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
See also
References
Further reading
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.