Rigorous justification for the space–split sensitivity algorithm to compute linear response in Anosov systems

Abstract

Ruelle (1997 Commun. Math. Phys. 187 227–41; 2003 Commun. Math. Phys. 234 185–90) (see also Jiang 2012 Ergod. Theor. Dynam. Syst. 32 1350–69) gave a formula for linear response of transitive Anosov diffeomorphisms. Recently, practically computable realizations of Ruelle’s formula have emerged that potentially enable sensitivity analysis of certain high-dimensional chaotic numerical simulations encountered in the applied sciences. In this paper, we provide full mathematical justification for the convergence of one such efficient computation, the space–split sensitivity, or S3, algorithm (Chandramoorthy and Wang 2022 SIAM J. Appl. Dyn. Syst. 21 735–81). In S3, Ruelle’s formula is computed as a sum of two terms obtained by decomposing the perturbation vector field into a coboundary and a remainder that is parallel to the unstable direction. Such a decomposition results in a splitting of Ruelle’s formula that is amenable to efficient computation. We prove the existence of the S3 decomposition and the convergence of the computations of both resulting components of Ruelle’s formula.

A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

“…For this setting one can on the one hand still verify hyperbolicity of the map by an invariant cone condition, while on the other hand a non-vanishing value for α avoids an additional inversion symmetry, which would occur if µ exp(i α) were real valued. Nevertheless, the following numerical results do not seem to depend on these considerations 6 . Since (1) relates the local expansion and contraction rates with the stable and unstable directions, the degree of smoothness of the hyperbolic splitting is mirrored by the smoothness of these rates.…”

“…We would also like to remark that the lack of higher-order regularity of statistical quantities pertaining to a smooth dynamical system is also of great import for applications. For example, the lack of differentiability of unstable Jacobians has posed an obstacle for the efficient computation of linear response using Ruelle's formula [29] (see [6,12,20,24] and [3, section 5.3] for background and a broader overview of linear response). It is worth noting that our choice of area-preserving Blaschke maps as perturbations is not well-suited to numerically study this particular problem (due to the perturbations all sharing Lebesgue measure as their relevant invariant measure, thus giving rise to vanishing derivatives).…”