I think you are asking the wrong question.  A Lyapunov function is used to determine if an equilibrium point (the origin,, without loss of generality), is (asymptotically) stable.  The Lyapunov exponent is a different thing. It measures the rate of variation of the distance between two trajectories.  It cannot be obtained analytically, but numerically.  See https://en.wikipedia.org/wiki/Lyapunov_exponent  for a definition.  Check http://sprott.physics.wisc.edu/chaos/lyapexp.htm to get an idea how to compute it.   Here is some code to compute it using Matlab. http://www.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode

Have a look on XPP/AUTH, has nice features to study chaos.  http://www.math.pitt.edu/~bard/bardware/tut/xppchaos.html

No method is available to construct Liapunov function.  You can construct Liapnunov function by trial and error method. 

There is no systematic approach to construct Lyapunov function candidate. However,  in many cases involving control of nonlinear systems, you can intuitively use the number of state variables of the system to construct it. Highly valued textbooks on Control System Design and Analysis by Slotine and Li  and that of Hassan Khalil would surely be of benefit to you in understanding Lyapunov stability principles and Lyapunov function

sylvester s conditions

Thanks Prabir Panja, Edwin Umoh.

Mr. Ahmed, please, what do you mean of "Sylvester s conditions"?

sylvesete`s condition the the rules which can be laipnunov function positive definite for linear and non-linear function

I think you are asking the wrong question.  A Lyapunov function is used to determine if an equilibrium point (the origin,, without loss of generality), is (asymptotically) stable.  The Lyapunov exponent is a different thing. It measures the rate of variation of the distance between two trajectories.  It cannot be obtained analytically, but numerically.  See https://en.wikipedia.org/wiki/Lyapunov_exponent  for a definition.  Check http://sprott.physics.wisc.edu/chaos/lyapexp.htm to get an idea how to compute it.   Here is some code to compute it using Matlab. http://www.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode

Have a look on XPP/AUTH, has nice features to study chaos.  http://www.math.pitt.edu/~bard/bardware/tut/xppchaos.html

Thanks a lot Luis, for the explanation 

No problem!