Validity of the Lyapunov exponent approximation I was trying to get the Lyapunov exponent for some dynamical nonlinear systems and found that it is not true (as I had expected) that the distance between two trajectories with slightly different initial conditions always grows exponentially (to a good approximation).
Instead I found (for the logistical equation and the Lorenz system) that the logarithmic plot of the distance between trajectories is constant at the start, then goes linear (i.e. exponential) and then goes back to constant.
Why is this? The only condition I've found for the Lyapunov approximation to work is that the system must be linearizable. Why does the approximation fail in some intervals of time?
 A: 
Instead I found (for the logistical equation and the Lorenz system) that the logarithmic plot of the distance between trajectories is constant at the start

This should not happen and I cannot confirm this. Here is the separation of two trajectories I get for the logistic map (averaged over 10000 realisations):

And here is the same for the Lorenz system:

The initial higher slope for the Lorenz system is likely due to an initial displacement near the separatrix, i.e., a region where non-linearity plays a role at very small length scales. See, e.g., Wolf et al. In fact, the Lorenz system is not the best system to start calculating Lyapunov exponents with the Wolf method.
Finally note that for many systems, it takes some time for the separation to align itself to the direction of largest growth, and until this has happened, the separation may grow slower or even shrink. Still, it should not be constant, and this does neither apply to the logistic map (as there is only one direction) nor to the Lorenz system (due to the effect mentioned in the last paragraph.

The only condition I've found for the Lyapunov approximation to work is that the system must be linearizable.

But the linearisability must hold on the length scale of the distance of trajectories. As the systems are non-linear, a length scale on which linearisability is not given anymore must inevitably exist. Often, this length scale corresponds to the size of the attractor, when trajectories begin to approach each other again due to the finite size of the attractor. However, for certain systems, it may be a smaller length scale. 
Note that more advanced methods of calculating the Lyapunov exponent (e.g., Benettin et al.’s) solve this problem by only working with infinitesimal separations.
A: The behavior is approximately constant near the beginning because at very short timescales, the transformation of the phase space induced by the time evolution is basically a deformation where the distances change by a factor of $O(1)$.
The middle phase is when the chaos is actually growing  and the distances are growing exponentially.
At the end, at very long timescales, the maximum possible distance is basically reached and equilibrium is reached so the distance has nowhere to grow further.
