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

@article{Chandramoorthy2020ACR, title={A computable realization of Ruelle's formula for linear response of statistics in chaotic systems}, author={Nisha Chandramoorthy and Qiqi Wang}, journal={arXiv: Dynamical Systems}, year={2020} }

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 ${\cal O}(1/\sqrt{N})$ 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…

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## References

SHOWING 1-10 OF 36 REFERENCES

### Ruelle's linear response formula, ensemble adjoint schemes and Lévy flights

- Environmental Science
- 2004

A traditional subject in statistical physics is the linear response of a molecular dynamical system to changes in an external forcing agency, e.g. the Ohmic response of an electrical conductor to an…

### Sensitivity Analysis of Chaotic Systems Using Unstable Periodic Orbits

- PhysicsSIAM J. Appl. Dyn. Syst.
- 2018

A well-behaved adjoint sensitivity technique for chaotic dynamical systems that arises from the specialisation of established variational techniques to the unstable periodic orbits of the system, with empirical observation that most orbits predict approximately the same sensitivity.

### Dynamical ensembles in stationary states

- Physics
- 1995

We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in…

### Probability density adjoint for sensitivity analysis of the Mean of Chaos

- PhysicsJ. Comput. Phys.
- 2014

### Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps: A Functional Approach

- Mathematics
- 2018

Transfer operators associated with a dynamical system T and a weight g are important tools to understand the statistical properties of T , under appropriate smoothness and hyperbolicity conditions.…

### Unstable periodic orbits and the dimensions of multifractal chaotic attractors.

- PhysicsPhysical review. A, General physics
- 1988

The idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings is pursued.

### Covariant Lyapunov vectors

- Mathematics
- 2013

Recent years have witnessed a growing interest in covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here, we review the basic results of…

### Response Operators for Markov Processes in a Finite State Space: Radius of Convergence and Link to the Response Theory for Axiom A Systems

- Mathematics
- 2015

Using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state Markov processes. The results can be used for studying empirically…

### Sensitivity computation of statistically stationary quantities in turbulent flows

- Computer ScienceAIAA Aviation 2019 Forum
- 2019

This work derives the S3 algorithm under simplifying assumptions on the dynamics and presents a numerical validation on a low-dimensional example of chaos, which is a Monte-Carlo approach to the chaotic sensitivity computation problem.

### Feasibility Analysis of Ensemble Sensitivity Computation in Turbulent Flows

- Computer ScienceAIAA Journal
- 2019

The feasibility of Lea-Allen-Haine ensemble sensitivity computations under optimistic mathematical assumptions on the flow dynamics is analyzed and upper bounds on the rate of convergence of the ES method are estimated in numerical simulations of turbulent flow.