A Trajectory-Driven Algorithm for Differentiating SRB Measures on Unstable Manifolds

Abstract

Sinai--Ruelle--Bowen (SRB) measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectory-driven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We apply the concept of measure preservation together with the chain rule on smooth manifolds. Due to the recursive one-step nature of our derivations, the proposed procedure is memory-efficient and can be naturally integrated with existing Monte Carlo schemes widely used in computational chaotic dynamics. We numerically show the exponential convergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.

Approximating linear response of physical chaos

“…The resulting boundary terms vanish as proven in [7], which implies that in all integral transformations of this type, the boundary integrals can be neglected. The reader is also referred to [31] for a detailed description of every step of this process and relevant numerical examples. The major implication of Eq.…”

“…21 indicates that we need c i , i = 0, 1, ..., m, their unstable derivatives b, and derivatives of the SRB measure represented by g. We acknowledge that the computation of the SRB measure gradient is agnostic to the presence of the center manifold. Using the measure preservation property and chain rule on smooth manifolds, one can derive exponentially converging recursive formulas for g. The reader is referred to the authors' previous work published in [31] for a detailed derivation and analysis of a trajectory-driven algorithm for g. Therefore, we only need to modify the way b is computed in the presence of the neutral subspace. Once b is found, the unstable part is computed similarly to its neutral counterpart, by summing up K k-time correlations.…”

“…21) involves only self-derivatives of c i , i.e., b i,j with i = j, we show that in order to find a trajectory-following recursion, we also need all possible cross-derivatives of c i . The main tool used in the derivation of these formulas is the measure-based parameterization of local unstable manifolds with orthonormal gradients [31]. It means that the m-dimensional unstable manifold U k including x k , i.e., the point of M crossed by the trajectory at the k-th time step, is parameterized as follows: x k (ξ) : [0, 1] m → U k ⊂ M such that x k (ξ) is the multivariate inverse cumulative distribution (quantile function) and ∇ ξ k x k = Q k .…”

“…The chart coordinates ξ k are updated stepby-step to ensure the orthogonality of the gradient ∇ ξ k x k = [∂ ξ 1 k x k , ..., ∂ ξ m k x k ]. A more rigorous description and analysis of this coordinate transformation can be found in [31].…”

On Some Aspects of the Response to Stochastic and Deterministic Forcings

“…Directly implementing the Ruelle response formulas is particularly nontrivial in the case of deterministic chaotic systems, because of the radically different properties of the tangent space in its unstable and stable directions [9]. Fortunately, recent contributions based on adjoint and shadowing methods seem to provide a convincing way forward: [10][11][12][13]. Ruelle's response theory has been then reformulated and extended through the viewpoint of functional analysis by studying via a perturbative expansion the corrections to the transfer operator [14] due to the applied forcing [15][16][17].…”