# Chaos Theory: Hénon Map

Before we go onto looking at Lyapunov exponents, a statistic which helps us determine whether a system is chaotic or not, we will be looking at 2-D maps. In particular we will be examining the Hénon map, a discrete time dynamical system. First introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model, it has become on the of most studied examples of systems that exhibit chaotic behaviour. The Hénon map takes $(x_n,y_n)$ to a new point by the recurrence relation described by

$x_{n+1}=y_n+1-ax_n^2, \qquad \qquad y_{n+1}=bx_n \qquad \qquad \qquad \qquad \qquad (3).$

The map is dependent on two parameters a and b. We can see that if we have $b=0$ then the map reduces to a quadratic map. The classical Hénon map, which has $a=1.4$ and $b=0.3$. For these values the map is chaotic, and the system resembles a boomerang shape as seen below. Known as the Hénon attractor; it has become another icon of chaos theory alongside the bifurcation diagram and Lorenz attractor.

Hénon map for $a=1.4$, $b=0.3$ and initial conditions $x_0=0.5$ and $y_0=0.5$

The Hénon attractor is a strange attractor, this is because the dimension of the attractor is non-integer and is usually associated with systems that are chaotic.

Sources:

HILBORN, R.C. (2000). Chaos and Nonlinear Dynamics: An introduction for Scientists and Engineers. UK. Oxford University Press.

PEITGEN, H.O., JURGENS, H. & SAUPE, D. (2004). Chaos and Fractals: New Frontiers of Science Second Edition, New York, USA. Springer.