Markov matrix perturbations to optimize dynamical and entropy functionals

An important problem in applied dynamical systems is to compute the external forcing that provokes the largest response of a desired observable quantity. For this, we investigate the perturbation theory of Markov matrices in connection with linear response theory in statistical physics. We use perturbative expansions to derive linear algorithms to optimize physically relevant quantities such as: entropy, Kullback-Liebler-divergence and entropy production of Markov matrices and their related probability vectors. These optimization algorithms are applied to Markov chain representations of discrete and continuous flows in and out of equilibrium. We consider Markov matrix representations originating from Ulam-type approximations of transfer operators and a reduced order model of a turbulent flow based on unstable periodic orbits theory. We also propose a numerical protocol to recast matrix perturbations into vector field perturbations. The results allow to physically interpret the obtained optimizing perturbations without knowledge of the underlying equations, in a data-driven way.

• Publication year

  2025

• Venue

  Unknown venue

• Publication date

  2025-07-18

• Fields of study

  Mathematics, Physics

• Identifiers
  arXiv 2507.14040
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Optimal response for stochastic differential equations by local kernel perturbations.

  2025cited by this paper
• Optimal linear response for Anosov diffeomorphisms

  2025cited by this paper
• Optimal response for hyperbolic systems by the fast adjoint response method

  2025cited by this paper
• Interpretable and Equation-Free Response Theory for Complex Systems

  2025cited by this paper
• A general framework for linking free and forced fluctuations via Koopmanism

  2025cited by this paper
• The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

  2024cited by this paper
• Bridging the Gap Between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response

  2024cited by this paper
• Optimal linear response for expanding circle maps

  2023cited by this paper
• Heterogeneity of the attractor of the Lorenz ’96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties

  2023cited by this paper
• Direct Path from Turbulence to Time-Periodic Solutions.

  2023cited by this paper
• Linear and fractional response for nonlinear dissipative SPDEs

  2022cited by this paper
• On some aspects of the response to stochastic and deterministic forcings

  2022cited by this paper
• Optimal Linear Response for Markov Hilbert–Schmidt Integral Operators and Stochastic Dynamical Systems

  2021cited by this paper
• Decomposing the dynamics of the Lorenz 1963 model using unstable periodic orbits: Averages, transitions, and quasi-invariant sets.

  2021cited by this paper
• Coarse Graining the State Space of a Turbulent Flow Using Periodic Orbits.

  2020influential reference
• Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator.

  2020cited by this paper
• Response and Sensitivity Using Markov Chains

  2019influential reference
• Sensitivity Analysis: Matrix Methods in Demography and Ecology

  2019cited by this paper
• Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory

  2019cited by this paper
• Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

  2018influential reference
• Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

  2017cited by this paper
• Response Operators for Markov Processes in a Finite State Space: Radius of Convergence and Link to the Response Theory for Axiom A Systems

  2015influential reference
• An early warning indicator for atmospheric blocking events using transfer operators.

  2015cited by this paper
• Numerical Solution of Stochastic Differential Equations

  2015cited by this paper
• Streamwise-localized solutions at the onset of turbulence in pipe flow.

  2012cited by this paper
• Ergodicity Coefficients Defined by Vector Norms

  2011influential reference
• Estimating Long-Term Behavior of Flows without Trajectory Integration: The Infinitesimal Generator Approach

  2011cited by this paper
• Disentangling multi-level systems: averaging, correlations and memory

  2011cited by this paper
• A simple framework to justify linear response theory

  2009cited by this paper
• Fluctuation-dissipation: Response theory in statistical physics

  2008cited by this paper
• Information transfer between dynamical system components.

  2005cited by this paper
• HA MILTONIAN SYSTEMS AND TRANSFORMATIONS IN HILBERT SPACE

  2005cited by this paper
• Relaxation Times of Markov Chains in Statistical Mechanics and Combinatorial Structures

  2004cited by this paper
• Stability and exponential convergence of continuous-time Markov chains

  2003influential reference
• A Rigorous ODE Solver and Smale’s 14th Problem

  2002cited by this paper
• Linear response of the Lorenz system.

  2002cited by this paper
• Spectrum of Ornstein-Uhlenbeck Operators in Lp Spaces with Respect to Invariant Measures

  2002cited by this paper
• Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst

  2001cited by this paper
• Positive transfer operators and decay of correlations

  2000cited by this paper
• One-parameter semigroups for linear evolution equations

  1999cited by this paper
• Approximating Physical Invariant Measures of Mixing Dynamical Systems

  1998influential reference
• Unstable periodic orbits and attractor of the barotropic ocean model

  1998cited by this paper
• Nonequilibrium statistical mechanics near equilibrium: computing higher-order terms

  1998cited by this paper
• Dynamical ensembles in stationary states

  1995cited by this paper
• Chaos, Fractals, and Noise

  1994cited by this paper
• Sensitivity of finite Markov chains under perturbation

  1993cited by this paper
• The Lorenz equations

  1992cited by this paper
• Invariant measurement of strange sets in terms of cycles.

  1988cited by this paper
• Explicit forms for ergodicity coefficients and spectrum localization

  1984cited by this paper
• Network theory of microscopic and macroscopic behavior of master equation systems

  1976cited by this paper
• Perturbation theory and finite Markov chains

  1968cited by this paper
• The fluctuation-dissipation theorem

  1966cited by this paper
• Problems in modern mathematics

  1964cited by this paper
• Deterministic nonperiodic flow

  1963cited by this paper
• Norms and exclusion theorems

  1960cited by this paper

• Linear Response and Optimal Fingerprinting for Nonautonomous Systems

  2026cites this paper