Program 6 - Lyapunov Exponents, Part 2

This program provides an overview of the Lyapunov exponent of a one-parameter family of scalar maps. For illustration purposes we have selected the family F(a,x)=ax(1-x), but the program can be adjusted to other families. The family is entered as the first item of the program. The second is the starting point of the orbit used to determine the Lyapunov exponent. The third is the number of iterates we discard. The fourth entry are the number of iterates used to evaluate the Lyapunov exponent. The fifth entry is the parameter, with the sixth and seventh entry being its initial and final value. The last entry is the step size by which the parameter is increased.

Program Code

DLyapPlot[f_,y0_,k_,n_,ai_,a1_,a2_,an_]:=ListPlot[
	Table[{
		ai,
		Module[{y,i},
			y = Nest[f,y0,k];
			Log[Product[Abs[f'[y=f[y],ai]],{i,n-k}]^(1/(n-k))]
		]},
	{ai,a1,a2,an}],
	PlotJoined->True,PlotStyle->{Thickness[0.001]}
];

Example

First define the function.

g[y_] = a*y (1 - y);

Now run the program.

DLyapPlot[g, .2, 30, 200, a, 3, 4, .01]