A computable realization of Ruelle's formula for linear response of statistics in chaotic systems
Abstract
We present a computable reformulation of Ruelle's linear response formula for chaotic systems. The new formula, called SpaceSplit Sensitivity or S3, achieves an error convergence of the order ${\cal O}(1/\sqrt{N})$ using $N$ phase points. The reformulation is based on splitting the overall sensitivity into that to stable and unstable components of the perturbation. The unstable contribution to the sensitivity is regularized using ergodic properties and the hyperbolic structure of the dynamics. Numerical examples of uniformly hyperbolic attractors are used to validate the S3 formula against a naïve finitedifference calculation; sensitivities match closely, with far fewer sample points required by S3.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.04117
 Bibcode:
 2020arXiv200204117C
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematical Physics;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 21 pages, 2 figures, submitted