Henon

The Henon is a type of map, specifically a discrete-time dynamical system, known for exhibiting chaotic behavior. It was first introduced by Michel Henon in 1976 as a simplified two-dimensional model to study the behavior of the Hénon-Heiles system, a classic example of a nearly integrable Hamiltonian system. The Henon map is defined by the following set of equations: x_{n+1} = 1 - a*x_n^2 + y_n and y_{n+1} = b*x_n. The standard parameters commonly used to observe its chaotic nature are a = 1.4 and b = 0.3.

When these parameters are applied, the Henon map demonstrates a sensitive dependence on initial conditions, a hallmark of chaos. Small differences in the starting point (x0, y0) can lead to drastically different trajectories over time. Despite its chaotic nature for certain parameter values, the Henon map also possesses regions of regular, predictable behavior and fixed points. Studying the bifurcations and the structure of the phase space (the set of all possible states of the system) for the Henon map has been a significant area of research in the field of dynamical systems and chaos theory. Its simplicity allows for numerical exploration and visualization of complex attractors, such as the famous Henon attractor.