  Lyapunov exponents characterize the chaotic nature of dynamical systems by
quantifying the growth rate of uncertainty associated with the imperfect
measurement of initial conditions. Finite-time estimates of the exponent,
however, experience fluctuations due to both the initial condition and the
stochastic nature of the dynamical path. The scale of these fluctuations is
governed by the Lyapunov susceptibility, the finiteness of which typically
provides a sufficient condition for the law of large numbers to apply. Here, we
obtain a formally exact expression for this susceptibility in terms of the
Ruelle dynamical zeta function for one-dimensional systems. We further show
that, for systems governed by sequences of random matrices, the cycle expansion
of the zeta function enables systematic computations of the Lyapunov
susceptibility and of its higher-moment generalizations. The method is here
applied to a class of dynamical models that maps to static disordered spin
chains with interactions stretching over a varying distance, and is tested
against Monte Carlo simulations.
