Optimal linear response for Anosov diffeomorphisms

TL;DR

<3-5 sentence high-level summary> The paper develops a rigorous framework to optimally steer the linear response of the SRB measure for 2D Anosov diffeomorphisms by infinitesimal perturbations. It derives an explicit, regular expression for the derivative of the transfer operator on Gouëzel–Liverani anisotropic spaces and shows the SRB response exists and is given by (I−L0)^{-1} dotL(dotT)(f0). The authors prove the objective of maximizing the SRB response to an observable is continuous and admits a unique optimal perturbation under a convex constraint, providing explicit Fourier-coefficient formulas via a Riesz representation. A fast Fourier-based numerical scheme with convergence guarantees (via Fejér kernel mollification) estimates the optimal perturbation, and two case studies demonstrate targeted local localization of SRB mass near a fixed point and a period-two orbit. The work has potential applications in climate science and other fields where controlled long-term dynamical behavior is desired with small, well-chosen perturbations.

Abstract

It is well known that an Anosov diffeomorphism $T$ enjoys linear response of its SRB measure with respect to infinitesimal perturbations $\dot{T}$. For a fixed observation function $c$, we develop a theory to optimise the response of the SRB-expectation of $c$. Our approach is based on the response of the transfer operator on the anisotropic Banach spaces of Gouëzel--Liverani. We prove that the optimising perturbation $\dot{T}$ is unique for non-degenerate response functions and provide explicit expressions for the Fourier coefficients of $\dot{T}$. We develop an efficient Fourier-based numerical scheme to approximate the optimal vector field $\dot{T}$, along with a proof of convergence. The utility of our approach is illustrated in two numerical examples, by localising SRB measures with small, optimally selected, perturbations.

Figures (4)

• Figure 1: Optimal vector field computed with $n = 32$ Fourier modes along each coordinate direction, using a Sobolev norm scaling factor $\gamma=0.02$. The red lines are a segment of the stable manifold of the fixed point at the origin. A uniform scaling of the length of the vectors in this figure has been made for improved visualisation.
• Figure 2: SRB measure estimate of $T_\delta = T_0 +\delta\cdot \dot{T}$ for some positive $\delta$, where the average norm of $\delta\cdot\dot{T}$ is approximately 2% of the domain diameter. Note the concentration of mass about the fixed point near the origin.
• Figure 3: Optimal vector field computed with $n=32$ Fourier modes along each coordinate direction, using a Sobolev norm scaling factor $\gamma=0.02$. The blue background is an estimate of the SRB measure of the nonlinear Anosov map $T_0$. A uniform scaling of the length of the vectors in this figure has been made for improved visualisation.
• Figure 4: Estimate of the SRB measure of $T_\delta = T_0 +\delta\cdot \dot{T}$ for some positive $\delta$, where the average norm of $\delta\cdot\dot{T}$ is approximately 1.2% of the domain diameter. Note the strong concentration of mass about the period-two orbit.

Theorems & Definitions (16)

• Proposition 2.1
• Theorem 3.1
• proof
• Theorem 3.2: Theorem 2 KL99
• Theorem 3.3
• proof
• Theorem 4.1
• proof
• Proposition 4.2
• Theorem 4.3