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.

H$\acute{e}$non map

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.

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