Efficient Computation of Linear Response of Chaotic Attractors with One-Dimensional Unstable Manifolds

The sensitivity of time averages in a chaotic system to an infinitesimal parameter perturbation grows exponentially with the averaging time. However, long-term or ensemble statistics often vary differentiably parameters. Ruelle's response theory gives rigorous formula for these parametric derivatives linear response. But direct evaluation this is ill-conditioned, and hence downstream applications analysis, such as optimization uncertainty quantification, have been computational challenge dynamical systems. This paper presents space-split (S3) algorithm transform into well-conditioned ergodic-averaging computation. We prove decomposition that differentiable on unstable manifold, which we assume be one-dimensional. ensures one resulting terms, stable contribution, can computed using regularized tangent equation, similarly nonchaotic system. remaining term, known converted efficiently computable ergodic average. In process, develop new algorithms, may useful beyond response, compute vector field direction. S3 algorithm, combines ingredients enter contributions, converges like Monte Carlo approximation formula. presented here first step toward full-fledged analysis systems, wherever limited due lack availability sensitivities.