Hénon map

In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map,[1] is a discrete-time dynamical system.

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.

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.

The Hénon map may be decomposed into the composition of three functions acting on the domain one after the other.

If one solves the one-dimensional Hénon map for the special case: One arrives at the simple quadradic: Or The quadratic formula yields: In the special case b=1, this is simplified to If, in addition, a is in the form

the formula is further simplified to In practice the starting point (X,X) will follow a 4-point loop in two dimensions passing through all quadrants.

In 1976 France, the Lorenz attractor is analyzed by the physicist Yves Pomeau who performs a series of numerical calculations with J.L.

[4] The analysis produces a kind of complement to the work of Ruelle (and Lanford) presented in 1975.

It is the Lorenz attractor, that is to say, the one corresponding to the original differential equations, and its geometric structure that interest them.

Pomeau and Ibanez combine their numerical calculations with the results of mathematical analysis, based on the use of Poincaré sections.

Stretching, folding, sensitivity to initial conditions are naturally brought in this context in connection with the Lorenz attractor.

If the analysis is ultimately very mathematical, Pomeau and Ibanez follow, in a sense, a physicist approach, experimenting with the Lorenz system numerically.

The importance will be revealed by Pomeau himself (and a collaborator, Paul Manneville) through the "scenario" of Intermittency, proposed in 1979.

The second path suggested by Pomeau and Ibanez is the idea of realizing dynamical systems even simpler than that of Lorenz, but having similar characteristics, and which would make it possible to prove more clearly "evidences" brought to light by numerical calculations.

Since the reasoning is based on Poincaré's section, he proposes to produce an application of the plane in itself, rather than a differential equation, imitating the behavior of Lorenz and its strange attractor.

In January 1976, Pomeau presented his work during a seminar given at the Côte d'Azur Observatory, attended by Michel Hénon.

Michel Hénon uses Pomeau’s suggestion to obtain a simple system with a strange attractor.

For general nonlinear systems, the eigenfunctions of this operator cannot be expressed in any nice form.

it can be shown that almost all initial conditions inside the unit sphere generate chaotic signals with largest Lyapunov exponent

One can generate, for example, band-limited chaotic signals using digital filters in the feedback loop of the system.

Hénon attractor for a = 1.4 and b = 0.3
Hénon attractor for a = 1.4 and b = 0.3
Orbit diagram for the Hénon map with b=0.3 . Higher density (darker) indicates increased probability of the variable x acquiring that value for the given value of a . Notice the satellite regions of chaos and periodicity around a=1.075 -- these can arise depending upon initial conditions for x and y .
Variation of 'b' showing the Bifurcation diagram. The boomerang shape is further drawn in bold at the top. Initial coordinates for each cross-section is (0, -0.2). Achieved using Python and Matplotlib.
Classical Hénon map (15 iterations). Sub-iterations calculated using three steps decomposition.
Hénon map in 4D. The range for b is -1.5 to 0.5 and for a it is -2.3 to 1.0. All planar cross-sections that in each image of the video are empty indicates that for those cross-sections, the points diverged to infinity and were not plotted.
An approximate Koopman mode of the Hénon map found with a basis of 50x50 Gaussians evenly spaced over the domain. The standard deviation of the Gaussians is 3/45 and a 100x100 grid of points was used to fit the mode. This mode has eigenvalue 0.998, and it is the closest to 1. Notably, the dark blue region is the stable manifold of strange attractor.