Function File: output = lyap_k (X)
Function File: output = lyap_k (X, paramName, paramValue, …)

Estimates the maximum Lyapunov exponent using the algorithm described by Kantz on the TISEAN reference page:

http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/docs/chaospaper/citation.html

Input

X

Must be realvector.

Parameters

mmax

Maximum embedding dimension to use [default = 2].

mmin

Minimum embedding dimension to use [default = 2].

d

Delay used [default = 1].

rlow

Minimum length scale to search neighbors [default = 1e-3].

rhigh

Maximum length scale to search neighbors [default = 1e-2].

ecount

Number of length scales to use [default = 5].

n

Reference points to use [all].

s

Number of iterations in time [default = 50].

t

’theiler window’ [default = 0].

Switch

verbose

Prints information about the current length scale at runtime.

Output

The output is a struct array of size:

'ecount' x ('mmax' - 'mmin' + 1)

It has the following fields:

  • eps - holds the epsilon for the exponent
  • dim - holds the embedding dimension used in exponent
  • exp - contains the exponent data. It consists of 3 columns:
    1. The number of the iteration
    2. The logarithm of the stretching factor (the slope is the Laypunov exponent if it is a straight line)
    3. The number of points for which a neighborhood with enough points was found

See also: demo lyap_k, lyap_r, lyap_spec.

Algorithms

The algorithms for this functions have been taken from the TISEAN package.

Demonstration 1

The following code

 idx    = (1:2500).';
 in     = sin (idx./360) + cos (idx ./ 180);
 mmax   = 20;
 res    = lyap_k (in, 'mmin',2,'mmax',mmax,'d',6,'s',400,'t',500);

 cla reset
 hold on
 for j=2:mmax
   plot (res(1,j-1).exp(:,1),res(1,j-1).exp(:,2),'r');
 endfor
 axis tight
 xlabel ("t [flow samples]");
 ylabel ("S(eps, embed, t)");
 hold off

Produces the following figure

Figure 1

Package: tisean