Hénon map

The Hénon map is a two-dimensional discrete-time dynamical system. It is one of the most studied examples of dynamical systems exhibiting chaotic behavior. It maps a point (x, y) in the plane to a new point using the following equations:

x_n+1 = 1 - ax_n² + y_n y_n+1 = bx_n

Here, a and b are parameters that control the behavior of the map. The parameter a controls the degree of nonlinearity of the system, while b represents the determinant of the Jacobian matrix and thus measures the rate of area contraction of the map. Typically, a is a positive value, and b is a small positive value (often set to 0.3).

The Hénon map is interesting because, for certain parameter values, it exhibits chaotic behavior, meaning that trajectories are sensitive to initial conditions and unpredictable in the long run. It is a simplified model that captures many features observed in more complex dynamical systems.

The map's dynamics can include periodic orbits, quasi-periodic orbits, and chaotic attractors, depending on the values of the parameters a and b. The Hénon attractor, a fractal structure, is a well-known example of a strange attractor produced by this map.

The map has found applications in various fields, including physics, biology, and economics, as a model for complex systems. Analyzing the Hénon map helps researchers understand the fundamental principles of chaotic dynamics.