An ergodic averaging method to differentiate covariant Lyapunov vectors

Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of CLVs along their own directions. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale-Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the CLV self-derivatives with a statistical linear response formula.Comment: 28 pages, 13 figures, under revie

Ergodic Sensitivity Analysis of One-Dimensional Chaotic Maps

Sensitivity analysis in chaotic dynamical systems is a challenging task from a computational point of view. In this work, we present a numerical investigation of a novel approach, known as the space-split sensitivity or S3 algorithm. The S3 algorithm is an ergodic-averaging method to differentiate statistics in ergodic, chaotic systems, rigorously based on the theory of hyperbolic dynamics. We illustrate S3 on one-dimensional chaotic maps, revealing its computational advantage over naive finite difference computations of the same statistical response. In addition, we provide an intuitive explanation of the key components of the S3 algorithm, including the density gradient function.Comment: 20 pages, 17 figure

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

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.Comment: 32 pages, 13 figures, submitted to journal, under revie

Implant Compression Necrosis: Current Understanding and Case Report

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/141362/1/jper0700.pd