Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems

• Published in: Communications in nonlinear science & numerical simulation 2021-10, Vol.101 (C), p.105906, Article 105906
• Main Authors: Śliwiak, Adam A., Chandramoorthy, Nisha, Wang, Qiqi
• Format: Article
• Language: English
• Subjects: Chaos theory; Chaotic dynamical systems; Density; Dynamical systems; Linear response theory; Numerical analysis; Parameter sensitivity; Sensitivity analysis; Simulation; SRB density gradient function; Statistics
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2021.105906

•A numerical method for assessing the validity of the linear response is proposed.•Linear response is applicable when the density gradient is Lebesgue-integrable.•The proposed criterion is validated in chaotic systems with 1D unstable manifolds.•An ergodic-averaging scheme to compute the density gradient is developed.•Examples of chaotic dynamics with smooth/rough parameter dependence are presented. 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.