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

Abstract

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

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