arXiv subjects

Stefano Galatolo

Publications and source records attributed to Stefano Galatolo.

At least 19 recordsLinked to original sources

arXiv2026

Ulam Approximation for Nonautonomous Systems: Equivariant Measures and Linear Response

Despite the prevalence of nonautonomous systems in applications, their statistical properties are much less understood than in the autonomous setting. Building on recent results on response theory for nonautonomous systems, we study the approximation of equivariant families and of their linear response by Ulam-type finite-dimensional reductions. First, we show that coarse-graining procedures associated with the classical Ulam method, and more generally with suitable finite-element projections, provide rigorous approximation of equivariant families for sequential systems with memory loss. Second, for systems whose transfer operators are regularizing, we prove that the linear response of the reduced finite-state Markov model converges to the projected linear response of the original system. To the best of our knowledge, a general approximation result of this type has not previously been established in this form, even in the autonomous case. We complement the analysis with numerical experiments on simple but representative time-dependent diffusive models. These results provide a rigorous foundation for the use of Markov approximations in the study of statistical properties of nonautonomous complex systems which almost invariably relies on finite-scale and finite-precision descriptions of their states and dynamics.

math.DS↗

arXiv2026

Optimal response for stochastic differential equations in $\mathbb{T}^d$ with perturbations on the drift term

We study stochastic differential equations on the $d$-dimensional flat torus $\mathbb{T}^d$ with drift and perturbation coefficients in $L^{\infty}(\mathbb{T}^d;\mathbb{R}^d)$ and additive non-degenerate noise. For the associated transfer operators, we analyse the dependence of the stationary measure and of the expectation of a given observable on small perturbations of the drift. In this framework, we prove a linear response formula for the invariant density and for the expectation of a given observable. We then address an optimal response problem, namely the determination of admissible perturbations that maximise the first-order variation of a prescribed observable. We establish existence of optimal perturbations and, in a Hilbert space framework, prove uniqueness and provide an explicit characterisation of the optimiser. This yields a practical Fourier-based numerical method, which we implement in several numerical examples, including both low and high-dimensional settings.

math.DS↗

arXiv2026

A Mathematical Framework for Linear Response Theory for Nonautonomous Systems

Linear response theory aims to predict how an additional forcing alters the statistical properties of a reference system. Such questions have been studied predominantly for autonomous dynamical systems, although many systems in the physical, natural, and social sciences are inherently nonautonomous and evolve under time-dependent external forcings. In this setting, one would like to understand how the system's time-dependent statistical properties change when an additional infinitesimal forcing is applied. Despite its practical relevance, this question has received a rigorous mathematical treatment only for a limited number of systems and perturbations. We develop a rigorous linear response theory for a broad class of deterministic and random nonautonomous systems under uniform assumptions extending those commonly used in the autonomous setting. A central ingredient is rapid loss of memory, namely sufficiently fast forgetting of initial conditions along the nonautonomous evolution. Our main strategy is to reformulate the sequential dynamics as a fixed-point problem for a global transfer operator acting on a sequence space of measures. This yields explicit causal response formulas for predicting the effect of small perturbations on time-dependent statistical states. We illustrate the theory for sequential compositions of expanding maps and for sequential compositions of random maps with additive noise, where uniform positivity of the noise implies exponential loss of memory. We also prove linear response for compact reflected Euler-Maruyama discretizations of dissipative nonautonomous stochastic differential equations. Finally, we apply the framework to a finite-dimensional stochastic discretization of the Ghil-Sellers energy balance model and study its response to a time-dependent perturbation of the greenhouse parameter.

math.DS↗

arXiv2026

Efficient computation of stationary measures and the Lyapunov Landscape for families random dynamical systems with smooth additive noise

We present an efficient and validated method for approximating the stationary measures of random dynamical systems with smooth additive noise. The approach leverages the strong regularizing properties of the associated transfer operator through a finite-dimensional reduction based on Fourier approximation. Explicit error bounds make the method suitable for use in computer-assisted proofs and rigorous numerical investigations; in particular, its efficiency {\em enables systematic explorations of parameter space}. The method provides access to the stationary measure and supports the analysis of key statistical properties of the system. As an application, we study noise-induced phenomena, focusing on the transition from positive to negative Lyapunov exponent (commonly known as Noise Induced Order) in families of random unimodal maps with Gaussian additive noise. By analyzing the Lyapunov exponent as a function of the system parameters, we identify transitions along a hypersurface in parameter space. The parameters we consider include the standard deviation (intensity) of the Gaussian noise and the shape of the unimodal map.

math.DS↗

arXiv2025

Stabilizing the Staking Rate, Dynamically Distributed Inflation and Delay Induced Oscillations

Dynamically distributed inflation is a common mechanism used to guide a blockchain's staking rate towards a desired equilibrium between network security and token liquidity. However, the high sensitivity of the annual percentage yield to changes in the staking rate, coupled with the inherent feedback delays in staker responses, can induce undesirable oscillations around this equilibrium. This paper investigates this instability phenomenon. We analyze the dynamics of inflation-based reward systems and propose a novel distribution model designed to stabilize the staking rate. Our solution effectively dampens oscillations, stabilizing the yield within a target staking range.

cs.CR↗

arXiv2025

Optimal response for stochastic differential equations by local kernel perturbations

We consider a random dynamical system on $\mathbb{R}^d$, whose dynamics is defined by a stochastic differential equation. The annealed transfer operator associated with such systems is a kernel operator. Given a set of feasible infinitesimal perturbations $P$ to this kernel, with support in a certain compact set, and a specified observable function $ϕ: \mathbb{R}^d \to \mathbb{R}$, we study which infinitesimal perturbation in $P$ produces the greatest change in expectation of $ϕ$. We establish conditions under which the optimal perturbation uniquely exists and present a numerical method to approximate the optimal infinitesimal kernel perturbation. Finally, we numerically illustrate our findings with concrete examples.

math.DS↗

arXiv2025

Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps

We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota-Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has a spectral gap (on which we have quantitative estimation). As an application we consider Lorenz-like two dimensional maps (piecewise hyperbolic with unbounded contraction and expansion rate): we prove that those systems have a spectral gap and we show a quantitative estimate for their statistical stability. Under deterministic perturbations of the system of size $δ$, the physical measure varies continuously, with a modulus of continuity $O(δ\log δ)$, which is asymptotically optimal for this kind of piecewise smooth maps.

math.DS↗

arXiv2025

Optimal Response for Hyperbolic Systems by the fast adjoint response method

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.

math.DS↗

arXiv2024

Stability of Fixed Points for Nonlinear Selfconsistent Transfer Operators via Cone Contractions

In this paper we investigate the action of self-consistent transfer operators (STOs) on Birkhoff cones and give sufficient conditions for stability of their fixed points. Our approach relies on the order preservation properties of STOs that can be established via the study of their differential. We focus on the study of STOs arising from strongly coupled maps both deterministic and noisy. Our approach allows for explicit estimates that we use to give examples of STOs with multiple stable fixed points some of which are shown to be far from the asymptotic behaviour of the corresponding system of finite coupled maps and give information only on long transients for the finite dimensional system.

math.DS↗

arXiv2024

A logarithm law for nonautonomous systems fastly converging to equilibrium and mean field coupled systems

We prove that if a nonautonomous system has in a certain sense a fast convergence to equilibrium (faster than any power law behavior) then the time $τ_{r}(x,y)$ needed for a typical point $x$ to enter for the first time in a ball $B(y,r)$ centered in $y$, with small radius \ $r $ scales as the local dimension of the equilibrium measure \ $μ$ at $y$, i.e. $$ \underset{r\rightarrow 0}{\lim }\frac{\log τ_{r}(x,y)}{-\log r}% =d_{μ}(y).$$ We then apply the general result to concrete systems of different kind, showing such a logarithm law for asymptotically authonomous solenoidal maps and mean field coupled expanding maps.

math.DS↗

arXiv2024

The differential of self-consistent transfer operators and the local convergence to equilibrium of mean field strongly coupled dynamical systems

We consider the differential of a self-consistent transfer operator at a fixed point of the operator itself and show that its spectral properties can be used to establish a kind of local exponential convergence to equilibrium: probability measures near the fixed point converge exponentially fast to the fixed point by the iteration of the transfer operator. This holds also in the strong coupling case. We also show that for mean field coupled systems satisfying uniformly a Lasota-Yorke inequality the differential does also. We present examples of application of the general results to self-consistent transfer operators based on deterministic expanding maps considered with different couplings, outside the weak coupling regime.

math.DS↗

arXiv2024

Linear response due to singularities

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a \emph{cusp} at the turning point, we recover the linear response. More precisely, let $T_\eps$ be a family of such cusp maps generated by changing the value of the turning point of $T_0$ by a deterministic perturbation and let $h_\eps$ be the corresponding invariant density. We prove that $\eps\mapsto h_\eps$ is differentiable in $L^1$ and provide a formula for its derivative.

math.DS↗

arXiv2023

Optimal linear response for expanding circle maps

We consider the problem of optimal linear response for deterministic expanding maps of the circle. To each infinitesimal perturbation $\dot{T}$ of a circle map $T$ we consider (i) the response of the expectation of an observation function and (ii) the response of isolated spectral points of the transfer operator of $T$. In each case, under mild conditions on the set of feasible perturbations $\dot{T}$ we show there is a unique optimal feasible infinitesimal perturbation $\dot{T}_{\rm optimal}$, maximising the increase of the expectation of the given observation function or maximising the increase of the spectral gap of the transfer operator associated to the system. We derive expressions for the unique maximiser $\dot{T}_{\rm optimal}$ in terms of its Fourier coefficients. We also devise a Fourier-based computational scheme and apply it to illustrate our theory.

math.DS↗

arXiv2023

A general framework for the rigorous computation of invariant densities and the coarse-fine strategy

In this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system trough a finite element reduction, by composing the associated transfer operator with a suitable finite dimensional projection (a discretization scheme) as in the well-known Ulam method. We introduce a general framework based on a list of properties (of the system and of the projection) that need to be verified so that we can take advantage of a so-called ``coarse-fine'' strategy. This strategy is a novel method in which we exploit information coming from a coarser approximation of the system to get useful information on a finer approximation, speeding up the computation. This coarse-fine strategy allows a precise estimation of invariant densities and also allows to estimate rigorously the speed of mixing of the system by the speed of mixing of a coarse approximation of it, which can easily be estimated by the computer. The estimates obtained here are rigourous, i.e., they come with exact error bounds that are guaranteed to hold and take into account both the discretiazation and the approximations induced by finite-precision arithmetic. We apply this framework to several discretization schemes and examples of invariant density computation from previous works, obtaining a remarkable reduction in computation time. We have implemented the numerical methods described here in the Julia programming language, and released our implementation publicly as a Julia package.

math.DS↗

arXiv2022

Statistical properties of dynamics. Introduction to the functional analytic approach

These are lecture notes for a simple minicourse approaching the satistical properties of a dynamical system by the study of the associated transfer operator (considered on a suitable functions or measures spaces). The following questions will be addressed: *existence of a regular invariant measure; *Lasota Yorke inequalities and spectral gap; *decay of correlations and some limit theorem; *stability under perturbations of the system *linear response *random systems *hyperbolic systems. The point of view taken is to present the general construction and ideas needed to obtain these results in the simplest way. For this, some theorem is proved in a form which is weaker than usually known, but with an elementary and simple proof. These notes are intended for the Hokkaido-Pisa University summer course 2021.

math.DS↗

arXiv2022

Self consistent transfer operators. Invariant measures, convergence to equilibrium, linear response and control of the statistical properties

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented in the paper are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We apply the general statements to examples of different nature: coupled continuous maps, coupled expanding maps, coupled systems with additive noise, systems made of \emph{different maps }coupled by a mean field interaction and other examples of self consistent transfer operators not coming from coupled maps. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.

math.DS↗

arXiv2021

Existence of multiple noise-induced transitions in a Lasota-Mackey map

We prove the existence of multiple noise-induced transitions in the Lasota-Mackey map, which is a class of one dimensional random dynamical system with additive noise. The result is achieved by the help of rigorous computer assisted estimates. We first approximate the stationary distribution of the random dynamical system and then compute certified error intervals for the Lyapunov exponent. We find that the sign of the Lyapunov exponent changes at least three times when increasing the noise amplitude. We also show numerical evidence that the standard non-rigorous numerical approximation by finite-time Lyapunov exponent is valid with our model for a sufficiently large number of iterations. Our method is expected to work for a broad class of nonlinear stochastic phenomena.

nlin.CD↗

arXiv2021

Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an $L^2$ kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subjected to additive noise. We develop a general setting in which these optimisation problems have a unique solution and construct explicit formulae for the unique optimal perturbations. We apply our results to a Pomeau-Manneville map and an interval exchange map, both subjected to additive noise, to explicitly compute the perturbations provoking maximal responses.

math.DS↗