A computable realization of Ruelle's formula for linear response of statistics in chaotic systems

Abstract

We present a computable reformulation of Ruelle's linear response formula for chaotic systems. The new formula, called Space-Split Sensitivity or S3, achieves an error convergence of the order $\mathcal{O}(1/\sqrt{N})$ using $N$ phase points. The reformulation is based on splitting the overall sensitivity into that to stable and unstable components of the perturbation. The unstable contribution to the sensitivity is regularized using ergodic properties and the hyperbolic structure of the dynamics. Numerical examples of uniformly hyperbolic attractors are used to validate the S3 formula against a naïve finite-difference calculation; sensitivities match closely, with far fewer sample points required by S3.

arXiv comment21 pages, 2 figures, submitted

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Cited by in corpus (11)

• On Some Aspects of the Response to Stochastic and Deterministic Forcings
• Response Theory and Phase Transitions for the Thermodynamic Limit of Interacting Identical Systems
• Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems
• Heterogeneity of the Attractor of the Lorenz '96 Model: Lyapunov Analysis, Unstable Periodic Orbits, and Shadowing Properties
• Variational optimization and data assimilation in chaotic time-delayed systems with automatic-differentiated shadowing sensitivity
• An ergodic averaging method to differentiate covariant Lyapunov vectors
• A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds
• Approximating linear response by nonintrusive shadowing algorithms
• Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems
• Fast differentiation of hyperbolic chaos
• Differentiating densities on smooth manifolds