If the flow is chaotic, then small initial errors, , in a 
trajectory will diverge exponentially.  
We are interested in calculating the stability—i.e., how fast do nearby 
trajectories diverge?
The Jacobi matrix of the velocity field, 
provides information about the local rate of divergence of 
nearby trajectories or the local rate of stretching of 
Lagrangian space.
We define the matrix H such that:
where I is the identity matrix.  It follows that:
The finite-time Lyapunov exponents are defined as the time average of the logarithms 
of the lengths of the principal components of the vector H over a time t:
where  is the ith Lyapunov exponent
of the system, while  is the ith principal component 
of the matrix H.
If we start with a set of orthonormal initial error vectors, 
 then the matrix H will map them to a set 
of final orthogonal error vectors of length .  
The action of the system maps an infinitesimal sphere of 
inititial points to an ellipsoid whose major axis is given by the 
 while the minor axis is given by , 
where N is the number of dimensions.
This definition of Lyapunov exponents is both more elegant and more appropriate
to real-world, continuous-time dynamical systems than the more usual definition based
on discrete function maps.
Chaos is defined as the existence of at least one positive Lyapunov exponent.
In a chaotic system, we call the Lyapunov exponent the asymptotic value of the greatest eigenvalue of H:
If there is any significant difference between the Lyapunov exponents then as an error vector evolves forward in time, any displacement in the direction of largest growth will tend to be magnified. Thus:
The Lyapunov exponent of a flow is a unique quantity, that characterizes the asymptotic separation of fluid particles in a given flow. It is often used as a measure of the 
efficiency of mixing, since it measures how fast trajectories separate from each other because of chaotic advection. The Lyapunov exponent can be computed by different methods:
* by following one single trajectory for very long times and computing .
* or by following an ensemble of trajectories for a given period of time, and computing the ensemble average: 
The equivalence of the two methods is due to the ergodicity of the chaotic system.