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 to a new point by the recurrence relation described by
The map is dependent on two parameters a and b. We can see that if we have then the map reduces to a quadratic map. The classical Hénon map, which has and . 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.
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.
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.