  In the current work we demonstrate the principal possibility of prediction of
the response of the largest Lyapunov exponent of a chaotic dynamical system to
a small constant forcing perturbation via a linearized relation, which is
computed entirely from the unperturbed dynamics. We derive the formal
representation of the corresponding linear response operator, which involves
the (computationally infeasible) infinite time limit. We then compute suitable
finite-time approximations of the corresponding linear response operator, and
compare its response predictions with actual, directly perturbed and measured,
responses of the largest Lyapunov exponent. The test dynamical system is a
20-variable Lorenz 96 model, run in weakly, moderately, and strongly chaotic
regimes. We observe that the linearized response prediction is a good
approximation for the moderately and strongly chaotic regimes, and less so in
the weakly chaotic regime due to intrinsic nonlinearity in the response of the
Lyapunov exponent, which the linearized approximation is incapable of
following.
