I recommend you to read the excellent survey by Charalampos Skokos http://arxiv.org/abs/0811.0882.

It depends on the problem. If you need to calculate only largest Lyapunov exponent, which is usually enough, you can use following link

http://sprott.physics.wisc.edu/chaos/lyapexp.htm

That algorithm is used in the paper

https://www.researchgate.net/publication/270824829_Farey_sequence_in_the_appearance_of_subharmonic_Shapiro_steps

Article Farey sequence in the appearance of subharmonic Shapiro steps

The answer is the the book Nonlinear Time Series Analysis.  Kants and Schreiber.

I imagine Taylor series for multivariable functions may be useful. Read my paper in my profile.

Dear Andrés Granados ,

What is the relationship  between Taylor series and Lyapunov exponent?!!!

If you put

delta Z(t)=exp(\lambda t) delta Z_o

In the Taylor series then you can find the vector \lambda up to the order you want. Numerical Taylor methods... suggest a procedure you may follow for this multivariable function. It is an idea. Apply chain rule to obtain the nonlinear system of equations on \lambda_i.

The Taylor series is for your ODE in the left side. The right side is the multiple polynomials in \lambda up to the grade you want to solve. Then in the limit   t \arrow 0 it becomes in a system of nonlinear equations in \lambda.

 See the draft

https://www.academia.edu/13605173/Lyapunov_Exponent_Calculation

@ Andrés Granados 

thank you sir for helping.

I actualized the references in the draft to complete the related information. If you need one of them only ask.

• Okay!

http://physionet.org/physiotools/lyapunov/

@ Darek Michalski

I was not able to download this file

Adnene;

try

http://physionet.org/physiotools/lyapunov/l1d2/

and also take a look at

http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/index.html

There is a well written paper by Wolf et al. to construct  the LLE from a time series

http://chaos.utexas.edu/manuscripts/1085774778.pdf

Thank you Dr. Darek Michalski and Dr. Daniil Yurchenko for your answers and suggestions.

I recommend you to read the excellent survey by Charalampos Skokos http://arxiv.org/abs/0811.0882.

Thanks for all giving valuable information

M.Berezowski, Spatio-temporal chaos in tubular chemical reactors with the recycle of mass.Chaos,Solitons&Fractals, 11, 1197-1204, 2000

http://www.sciencedirect.com/science/article/pii/S0960077999000260

As for calculating the Lyapunov exponent from the time series (without knowing the exact form of the underlined system behind), I suggest to refer to the algorithms developed in the seminal paper by A. Wlof et al., published in Physica D (1985, 16, pp. 285).    As for calculating the exponent from a deterministic dynamical systems, one can directly compute it according to the typical definition, which could be found through the Wikipedia.  

This last paper with its link was reccomended before by Daniil Yurchenko.

you can follow the link below

http://www.mathworks.com/matlabcentral/fileexchange/48084-lyapunov-exponent-estimation-from-a-time-series

If you are looking for algorithms to approximate Lyapunov exponents of nonlinear differential equations, other than those already recommended, you can read 

1. Luca Dieci, Jacobian free computation of Lyapunov exponents.

2. K. Ramasubramanian, M.S. Sriram, A comparative study of computation of Lyapunov spectra with different algorithms

The papers are (free) on Google.

Thank's 

For what values of the Lyapunov exponents the phase trajectory will be chaotic??!!

If at least one Lyapunov exponent ( in fact the maximum Lyapunov exponent) is positive then the trajectory is chaotic.

Is it possible to detect the Chaos without calculating the Lyapunov exponent. For example from the graph of the phase trajectory?

Lots of people just plot a projection of the trajectory in some 2-D space, and if it looks scrambled, they declare that the system is chaotic, and they are usually correct. However, if you do this, iterate the equations for quite a while before you begin plotting the results since chaotic transients are quite common for systems that eventually attract to a limit cycle or stable equilibrium point. If you plan to publish your claim of chaos in a reputable journal, you really need to show that the largest Lyapunov exponent is positive and statistically significant.