Journal
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
Volume 101, Issue -, Pages -Publisher
ELSEVIER
DOI: 10.1016/j.cnsns.2021.105906
Keywords
Chaotic dynamical systems; Sensitivity analysis; Linear response theory; SRB density gradient function
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Funding
- Air Force Office of Scientific Research [FA8650-19-C-2207]
- U.S. Department of Energy [DEFOA00020680018]
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This paper introduces a numerical procedure to assess the differentiability of statistics with respect to parameters in chaotic systems, based on the Lebesgue-integrability of a density gradient function, and develops a recursive formula for computing the density gradient, demonstrating its use in determining the differentiability of statistics.
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. (c) 2021 Elsevier B.V. All rights reserved.
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