Measuring dependency of chaotic dynamics on initial values by Lyapunov exponent, an original trajectory along with a perturbed one is needed. However, in a time series in which we face only one set of points (not perturbed one), how could it be possible to compute the Lyapunov exponent?
please refer to this paper
Barahona, M. and C.-S. Poon. Detection of nonlinear dynamics in short, noisy time series. Nature, 381:215-217 1996.
Http://www.physicsforums.com/showthread.php?t=475930
http://ideas.repec.org/c/boc/bocode/t741502.html
http://www.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode
Please, give a look to the R packages: fNonlinear, tseriesChaos (and the references therein)
You can compute the maximal Lyapunov exponent. using the local divergence rates of nearby trajectories.
For example, H. Kantz proposed the method (H. Kantz, "A robust method to estimate the maximal Lyapunov exponent of a time series", Phys. Lett. A, 1994. )
You can check this:
http://kops.ub.uni-konstanz.de/bitstream/handle/urn:nbn:de:bsz:352-opus-64213/Elbert_1995_A_Practical.pdf?sequence=1
or tihs:
http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.1/index.html
The last provides an opensource software. Forget matlab (if you drive a car it does not necessary means you understand how the car works).
if you use MatLab in addition you can apply the following m-code http://www.mathworks.com/matlabcentral/fileexchange/233.
Short theoretical background: http://mpej.unige.ch/~eckmann/ps_files/eckmannkamphorstruelle.pdf; https://www.math.lsu.edu/courses/capstone_course/fall2006/Lyapunovexp/background
I strongly recommend TISEAN 2.1, which is an open source software; it needs some initial workout. Nevertheless, it is a very useful package. See the link: http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.1/index.html
Tisean project is a good program. You can use either of Kantz or Rosetein algorithm with ease.
This is probably the first algorithm implemented and published for computing Lyapunov exponents from time series: http://www.sciencedirect.com/science/article/pii/0167278985900119
The Wolf, et al. paper cited by John Pearson is very readable and describes how one finds suitable points in the time series to use as "initial conditions" and how the LE is calculated from the separation of these points in time. It also contains FORTRAN code for implementing the method. Their method has been criticized because it assumes and does not test for exponential divergence of the orbits, and thus it cannot be used to distinguish chaos from noise. The Wolf algorithm is included in my commercial software, Chaos Data Analyzer (http://sprott.physics.wisc.edu/cda.htm), but I caution against its blind use or any of the other methods that purport to calculate a Lyapunov exponent from time-series data since the whole business of nonlinear time-series analysis is more of an art than a science.
I suggest using TISEAN package, which is a nonlinear time series analysis tool. Here's a link to the home page: https://www.pks.mpg.de/~tisean/Tisean_3.0.1/index.html
There's a command lyap_k (https://www.pks.mpg.de/~tisean/Tisean_3.0.1/index.html) which you can use to get the largest Lyapunov exponent if you have the data.
Have a look at https://www.youtube.com/watch?v=YL4twBVKNK0
https://au.mathworks.com/matlabcentral/fileexchange/48084-wolf-lyapunov-exponent-estimation-from-a-time-series
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