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Michele Gianfelice

Publications and source records attributed to Michele Gianfelice.

10 recordsLinked to original sources
arXiv2024

A note on the topological synchronisation of unimodal maps

In this note we complete the analysis carried on in \cite{CGSV} about the topological synchronisation of unimodal maps of the interval coupled in a master-slave configuration, by answering to the questions raised in that paper. Namely, we compute the weak limits of the invariant measure of the coupled system as the coupling strength $k\in\left( 0,1\right) $ tends to $0$ and to $1$ and discuss the uniqueness of the invariant measure of its random dynamical system counterpart, proving that the convergence of the associated Markov chain to its unique stationary measure is geometric. [CGSV] Caby Th., Gianfelice M., Saussol B., Vaienti S. "Topological synchronisation or a simple attractor?" Nonlinearity Vol. 36, no. 7, pp. 3603-3621 (2023).

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arXiv2024

On the generalized dimensions of physical measures of chaotic flows

We prove that if $μ$ is the physical measure of a $C^2$ flow in $\mathbb{R}^d, d \geq 3,$ diffeomorphically conjugated to a suspension flow based on a Poincaré application $R$ with physical measure $μ_{R}$, then $D_{q}(μ)=D_{q}(μ_{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q \neq1$. We also show that a similar result holds for the local dimensions of $μ$ and, under the additional hypothesis of exact-dimensionality of $μ_{R}$, that our result extends to the case $q=1$. We apply these results to estimate the $D_{q}$ spectrum associated with Rössler systems and turn our attention to Lorenz-like flows, proving the existence of their information dimension and giving a lower bound for their generalized dimensions.

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arXiv2023

Stochastic and statistical stability of the classical Lorenz flow under perturbations modeling anthropogenic type forcing

We review the results obtained in [GMPV] and [GV] on the stochastic and statistical stability of the classical Lorenz flow, where, looking at the Lorenz'63 ODE system as a simple - yet non trivial - model of the atmospheric circulation, the perturbation schemes introduced in these papers are designed to represent the effect of the so called anthropogenic forcing on the dynamics of the atmosphere.

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arXiv2022

Topological synchronisation or a simple attractor?

A few recent papers introduced the concept of topological synchronisation. We refer in particular to \cite{TS}, where the theory was illustrated by means of a skew product system, coupling two logistic maps. In this case, we show that the topological synchronisation could be easily explained as the birth of an attractor for increasing values of the coupling strength and the mutual convergence of two marginal empirical measures. Numerical computations based on a careful analysis of the Lyapunov exponents suggest that the attractor supports an absolutely continuous physical measure (acpm). We finally show that for some unimodal maps such acpm exhibit a multifractal structure.

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arXiv2020

Stochastic stability of the classical Lorenz flow under impulsive type forcing

We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in \cite{Ki}, \cite{Ke}, \cite{Me}. Namely, given a cross-section $\mathcal{M}$ for the unperturbed flow, each time the trajectory of the system crosses $\mathcal{M}$ the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the umperturbed flow is then carryed on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes.

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arXiv2020

Uniform bound of the entanglement for the ground state of the quantum Ising model with large transverse magnetic field

We consider the ground state of the quantum Ising model with transverse field $h$ in one dimension in a finite volume \[ Λ{_{m}:=\{-m,-m+1,\ldots,m+L\}\ .}% \] For $h$ sufficiently large we prove a bound for the entanglement of the interval $Λ_{0}:=\left\{ 0,..,L\right\} $ relative to its complement $Λ_{m}\backslashΛ_{0}$ which is uniform in $m$ and $L$. The bound is established by means of a suitable cluster expansion.

math-ph↗
arXiv2014

Dynamics And Kinetic Limit For A System Of Noiseless D-Dimensional Vicsek-Type Particles

We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles with a kinematic constraint on the velocities inspired by the original Vicsek's one \cite{VCB-JCS}. Interactions among particles are specified by a pairwise potential in such a way that the velocity of any given particle is updated to the weighted average velocity of all those particles interacting with it. The weights are given in terms of the interaction rate function. When the size of the system is fixed, we show the existence of an invariant manifold in the phase space and prove its exponential asymptotic stability. In the kinetic limit we show that the particle density satisfies a nonlinear kinetic equation of Vlasov type, under suitable conditions on the interaction. We study the qualitative behaviour of the solution and we show that the Boltzmann-Vlasov entropy is strictly decreasing in time.

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arXiv2011

On the recurrence and robust properties of Lorenz'63 model

Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.

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