Lyapunov Exponents - Demo 1

Henon Map

The Henon Map was investigated in demonstration 5.1 For a=1.4 b=0.3 this map shows chaotic behavior. One Lyapunov exponent is positive, corresponding to the sensitivity to initial conditions, and one negative, corresponding to the collapse onto the non-space-filling attractor.

You can check that the sum of the exponents is equal to log b. The convergence of the calculation can be quite slow. Convergence is improved by increasing the transient time over which the tangent vectors converge to the "Lyapunov eigenvectors" before the calculation of the magnification of the norm is started. The accuracy is increased by accumulating more points (increase Points and Speed for the iteration). You can also check that the values do not depend on the initial condition, by starting from a new random initial condition using the Reset button or clicking outside the stopped plot.

Decreasing a to 0.95 leads to a period 4 orbit, and now both exponents are negative, corresponding to the stability of this orbit.

You can study how the largest Lyapunov exponent increases towards zero as a is increased towards the value (around 1.025) where the period 4 cycle becomes unstable.

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Last modified Sunday, January 9, 2000
Michael Cross