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Isaia Nisoli

Publications and source records attributed to Isaia Nisoli.

At least 19 recordsLinked to original sources
arXiv2026

(co)Quasi-irreducible and (co)expanding random maps

We study quasi-irreducibility, expansion, and their cotangent duals for random $C^1$ (local) diffeomorphisms on compact invariant sets of a Riemannian manifold. Quasi-irreducibility is defined through stationary lifts to the projective tangent bundle: every lift must realize the Furstenberg--Kifer formula for the top Lyapunov exponent. We prove that, for ergodic stationary measures, this is equivalent to the absence of equators, namely non-random invariant subbundles on which the top exponent drops. Under simplicity of the first Lyapunov exponent for every stationary measure, quasi-irreducibility is also equivalent to vertical mostly contraction, contraction on average, a vertical spectral gap, and uniqueness of stationary projective lifts. We then apply these criteria to continuity of the top Lyapunov exponent and to expansion on average. In particular, expansion is characterized by positivity of the top Lyapunov exponent on all non-random invariant subbundles; under quasi-irreducibility, it is equivalent to positivity of the top Lyapunov exponent for every stationary measure. Dual statements hold for coquasi-irreducibility, coequators, coexpansion, and continuity of the bottom Lyapunov exponent. We also describe the interplay between expansion and coexpansion and extend the formalism to Grassmannian bundles, obtaining higher-dimensional versions controlling intermediate sums of Lyapunov exponents.

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arXiv2026

Integral representation of Lyapunov exponents

We introduce a new operator-theoretic construction of Lyapunov growth for Markov-driven systems. The construction is based on an abstract variational principle for asymptotic growth rates arising from a subadditive process generated by an intertwining pair of Markov operators on a measurable bundle with compact fibers: for each invariant base measure, the fiberwise-maximal growth rate equals the supremum of the fiber integral over invariant lifts, and this supremum is attained on an ergodic lift. For random linear bundle morphisms driven by Markovian place-dependent noise, this yields Lyapunov exponents defined intrinsically from the initial law on the state-phase space. We prove that these exponents coincide with the classical path-space skew-product Lyapunov spectrum, showing that the pointwise exponents depend only on the current noise state and initial position, not on the full noise realization. We also obtain new asymptotic formulas for the sums of Lyapunov exponents as conditional annealed growth rates along individual directions. As further consequences of this variational principle, we recover and extend to singular linear bundle morphisms the classical projective variational formulas.

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arXiv2026

Statistical periodicity in noise-induced order from Ruelle-Pollicott resonances

Noise-induced order (NIO) is a paradigmatic example of nontrivial noise-induced phenomena, characterized by pronounced spectral peaks and pseudoperiodic dynamics. We show that this periodicity is governed by the Ruelle-Pollicott resonances of the annealed transfer operator, independently of dynamical stability. Exact results for an analytically solvable model and numerical results for a modified Lasota--Mackey map show excellent agreement between resonance-based predictions and empirical power spectra across a broad range of noise amplitudes. Independent transitions in stability, diagnosed by the Lyapunov exponent, and statistical periodicity, diagnosed by the Ruelle-Pollicott resonances, give rise to three distinct types of NIO.

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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.

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arXiv2026

A pseudospectral approach to rigorous numerical estimation of resonances of transfer operators

{Ruelle-Pollicott} resonances, isolated eigenvalues of a transfer operator acting on suitably chosen Banach spaces, play a fundamental role in understanding the statistical properties of chaotic dynamical systems. In this paper, we introduce a pseudospectral approach, inspired by Householder's theorem, for the rigorous, computer-assisted estimation of resonances, providing regions where resonances must exist and precluding the presence of resonances elsewhere. The approach is general, and applies to the transfer operators of a wide variety of chaotic systems, including Anosov/ Axiom A diffeomorphisms and piecewise expanding maps. We implement this approach computationally for a class of analytic uniformly expanding maps of the circle. We anticipate that the pseudospectral framework developed here will be broadly applicable to other spectral problems in dynamical systems and beyond.

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arXiv2026

Certified spectral approximation of transfer operators and the Gauss map

We prove that the full spectral picture of a transfer operator (every isolated eigenvalue, eigenvector, and Riesz projector outside the essential spectral radius) can be approximated to arbitrary precision by finite-rank discretizations, with no spectral pollution. The method is a~posteriori: once a computable approximation bound is available (from compactness or a Doeblin--Fortet--Lasota--Yorke inequality, which may require hyperbolicity constants and adapted Banach spaces), the spectral gap and multiplicity are certified from computed data via a resolvent perturbation bound, with no further dynamical input. This applies both to compact operators on a single Banach space and to quasi-compact operators satisfying a Doeblin--Fortet--Lasota--Yorke inequality, extending Li's resolution of the Ulam conjecture from the invariant density to the entire discrete spectrum with certified error bounds at every finite truncation. As a benchmark, we certify the first $50$ nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator to at least $90$ rigorous decimal digits, together with their eigenvectors, Riesz projectors, and spectral gap, yielding a certified spectral expansion for Gauss--Kuzmin distributions with explicit error bounds and providing a rigorous answer to the Gauss--Babenko--Knuth problem on the spectral data of the Gauss map.

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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.

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arXiv2024

Designing robust trajectories by lobe dynamics in low-dimensional Hamiltonian systems

Modern space missions with uncrewed spacecraft require robust trajectory design to connect multiple chaotic orbits by small controls. To address this issue, we propose a control scheme to design robust trajectories by leveraging a geometrical structure in chaotic zones, known as a {\it lobe}. Our scheme shows that appropriately selected lobes reveal possible paths to traverse chaotic zones in a short time. The effectiveness of our method is demonstrated through trajectory design in both the standard map and Hill's equation.

nlin.CD↗
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.

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arXiv2022

How does noise induce order?

In this paper we present a general result with an easily checkable condition that ensures a transition from chaotic regime to regular regime in random dynamical systems with additive noise. We show how this result applies to a prototypical family of nonuniformly expanding one dimensional dynamical systems, showing the main mathematical phenomenon behind Noise Induced Order. Accepted at Journal of Statistical Physics

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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

Noise induced order for skew-products over a non-uniformly expanding base

Noise-induced order is the phenomenon by which the chaotic regime of a deterministic system is destroyed in the presence of noise. In this manuscript, we establish noise-induced order for a natural class of systems of dimension $\geq 2$ consisting of a fiber-contracting skew product a over nonuniformly-expanding 1-dimensional system.

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arXiv2019

Existence of Noise Induced Order, a Computer Aided Proof

We prove the existence of Noise Induced Order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belosouv-Zabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certified approximation of the stationary measure in the $L^{1}$ norm. This is realized by explicit functional analytic estimates working together with an efficient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure of the system varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a Hölder way when the noise amplitude increases.

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arXiv2017

A Rigorous Computational Approach to Linear Response

We present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the $L^{\infty}$-norm, the response of expanding circle maps under stochastic and deterministic perturbations. Moreover, we present an example where we compute, up to a pre-specified error in the $L^1$-norm, the response of the intermittent family at the boundary; i.e., when the unperturbed system is the doubling map.

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arXiv2015

Towards a Morse theory on Banach spaces via ultrafunctions

Morse Theory on Banach spaces would be a useful tool in nonlinear analysis but its development is hindered by many technical problems. In this paper we present an approach based on a new notion of generalized functions called \textquotedblleft ultrafunctions\textquotedblright\ which solves some of the technical questions involved.

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