Optimal Response for Hyperbolic Systems by the fast adjoint response method

TL;DR

This work develops a rigorous framework for the Optimal Response problem in uniformly hyperbolic systems, using the fast adjoint response formula to show the linear response operator $\mathcal{R}$ is bounded on $C^{1,\alpha}$ and extendable to $C^3$. Under convex feasibility sets $P$ in a Hilbert space $\mathcal{H}$, there exists a unique optimal perturbation $X_{opt}=v/\|v\|_{\mathcal{H}}$, where $v$ is the $\mathcal{H}$-representer of $R$, and the coefficients $c_i=R(b_i)$ give a practical Fourier-based construction on the torus. For toroidal phase spaces, $v$ is computable via Fourier modes in Sobolev spaces $H^p$, enabling an efficient, orbit-based algorithm that scales to high dimensions; for non-torus spaces, a PDE approach $A v=R$ provides a path to obtain the representer. The authors validate the method with numerical experiments in 2D, 3D, and 21D solenoid-like maps, demonstrating accurate recovery of the optimal perturbation and robustness to high dimensionality, with clear guidance on error sources and computational resources.

Abstract

In a uniformly hyperbolic system, we consider the problem of finding the optimal infinitesimal perturbation to apply to the system, from a certain set $P$ of feasible ones, to maximally increase the expectation of a given observation function. We perturb the system both by composing with a diffeomorphism near the identity or by adding a deterministic perturbation to the dynamics. In both cases, using the fast adjoint response formula, we show that the linear response operator, which associates the response of the expectation to the perturbation on the dynamics, is bounded in terms of the $C^{1,α}$ norm of the perturbation. Under the assumption that $P$ is a strictly convex, closed subset of a Hilbert space $\cH$ that can be continuously mapped in the space of $C^3$ vector fields on our phase space, we show that there is a unique optimal perturbation in $P$ that maximizes the increase of the given observation function. Furthermore since the response operator is represented by a certain element $v$ of $\cH$, when the feasible set $P$ is the unit ball of $\cH$, the optimal perturbation is $v/||v||_{\cH}$. We also show how to compute the Fourier expansion $v$ in different cases. Our approach can work even on high dimensional systems. We demonstrate our method on numerical examples in dimensions 2, 3, and 21.

Figures (10)

• Figure 1: The scatter plot of a typical orbit.
• Figure 2: Contour plot of $||B^j_{\vec{n}}||^2_{H^5}$ in log scale.
• Figure 3: Contour plot of $C^j_{\vec{n}} = R'( \tilde{B}^j_{\vec{n}})$. Left: $j=1$. Right: $j=2$.
• Figure 4: Vector field plot of the optimal perturbation $\frac{1}{4} X'_{opt}$.
• Figure 5: Linear responses and $\mu_\gamma(\Phi)$ of different perturbations. The dots are $\mu_\gamma(\Phi)$ for $f+\gamma X'_{opt}$ (indicated by red circles), $f+\gamma \tilde{B}^2_{(0,3)}$ (blue squares), and $f+\gamma \tilde{B}^2_{(14,14)}$ (black triangles). The short lines represent the linear responses computed by the fast response algorithm.

Theorems & Definitions (13)

• lemma 1
• Remark
• proof
• proposition 1
• Remark
• proof
• Definition 1: Strictly convex set
• proposition 2
• proof
• proposition 3