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Lyapunov exponents characterize the chaotic nature of dynamical systems byquantifying the growth rate of uncertainty associated with the imperfectmeasurement of initial conditions. Finite-time estimates of the exponent,however, experience fluctuations due to both the initial condition and thestochastic nature of the dynamical path. The scale of these fluctuations isgoverned by the Lyapunov susceptibility, the finiteness of which typicallyprovides a sufficient condition for the law of large numbers to apply. Here, weobtain a formally exact expression for this susceptibility in terms of theRuelle dynamical zeta function for one-dimensional systems. We further showthat, for systems governed by sequences of random matrices, the cycle expansionof the zeta function enables systematic computations of the Lyapunovsusceptibility and of its higher-moment generalizations. The method is hereapplied to a class of dynamical models that maps to static disordered spinchains with interactions stretching over a varying distance, and is testedagainst Monte Carlo simulations.

Cycle-expansion method for the Lyapunov exponent, susceptibility, and higher moments