Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems
arXiv:2101.08379 · doi:10.1016/j.cnsns.2021.105906
Abstract
An assumption of smooth response to small parameter changes, of statistics or long-time averages of a chaotic system, is generally made in the field of sensitivity analysis, and the parametric derivatives of statistical quantities are critically used in science and engineering. In this paper, we propose a numerical procedure to assess the differentiability of statistics with respect to parameters in chaotic systems. We numerically show that the existence of the derivative depends on the Lebesgue-integrability of a certain density gradient function, which we define as the derivative of logarithmic SRB density along the unstable manifold. We develop a recursive formula for the density gradient that can be efficiently computed along trajectories, and demonstrate its use in determining the differentiability of statistics. Our numerical procedure is illustrated on low-dimensional chaotic systems whose statistics exhibit both smooth and rough regions in parameter space.
arXiv comment32 pages, 13 figures, submitted to journal, under review
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- Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems
- A kind of Lagrangian chaotic property of the Arnold-Beltrami-Childress flow
- A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds
- Differentiating densities on smooth manifolds
- Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension