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Papers by Stefano Galatolo

$Research paper thumbnail of Multifractals via recurrence times?$

This letter is a comment on an article by T.C. Halsey and M.H. Jensen in Nature about using recur... more This letter is a comment on an article by T.C. Halsey and M.H. Jensen in Nature about using recurrence times as a reliable tool to estimate multifractal dimensions of strange attractors. Our aim is to emphasize that in the recent mathematical literature (not cited by these authors), there are positive as well as negative results about the use of such techniques. Thus one may be careful in using this tool in practical situations (experimental data).

Chaos, Solitons & Fractals, 2002

... AIC was first used in the context of dynamical systems by Brudno [2]. He defined a notion of ... more ... AIC was first used in the context of dynamical systems by Brudno [2]. He defined a notion of orbit complexity which is a measure of the quantity of information necessary to describe the orbit. ... First, the dictionary is empty and the ℓ-block is not parsed. ...

Dielectric Elastomers as Electromechanical Transducers, 2008

$Research paper thumbnail of Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps$

Ergodic Theory and Dynamical Systems, 2015

We consider a class of piecewise hyperbolic maps from the unit square to itself preserving a cont... more We consider a class of piecewise hyperbolic maps from the unit square to itself preserving a contracting foliation and inducing a piecewise expanding quotient map, with infinite derivative (like the first return maps of Lorenz like flows). We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error, with respect to the Wasserstein distance. We apply this to the rigorous computation of the dimension of the measure. We present a rigorous implementation of the algorithms using interval arithmetics, and the result of the computation on a nontrivial example of Lorenz like map and its attractor, obtaining a statement on its local dimension.

SIAM Journal on Applied Dynamical Systems, 2014

We describe a framework in which is possible to develop and implement algorithms for the approxim... more We describe a framework in which is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation.

Lecture Notes in Computer Science, 2006

ABSTRACT The study of dynamical systems suggests that an important indicator for their classifica... more ABSTRACT The study of dynamical systems suggests that an important indicator for their classification is the quantity of information that is needed to describe their orbits (here the orbits are transated into symbolic sequences by some geometrical construction). This leads to a definition of a complexity of a single orbit. This notion is flexible enough to give a refinement of the classical definition of entropy of a system. This concept is particularly interesting for systems with zero entropy (for motivations and some results see [S. Galatolo, Complexity, initial condition sensitivity, dimension and weak chaos in dynamical systems, Nonlinearity 16 (2003) 1219–1238]).

The study of dynamical systems suggests that an important indicator for their classification is t... more The study of dynamical systems suggests that an important indicator for their classification is the quantity of information that is needed to describe their orbits (here the orbits are transated into symbolic sequences by some geometrical construction). This leads to a definition of a complexity of a single orbit. This notion is flexible enough to give a refinement of the classical definition of entropy of a system. This concept is particularly interesting for systems with zero entropy (for motivations and some results see ).

We prove a quantitative recurrence result which allow to estimate the speed of approaching of a g... more We prove a quantitative recurrence result which allow to estimate the speed of approaching of a generic orbit to the discontinuities of a map. This result is applied to the study of complexity indicators for individual orbits generated by a certain zero-entropy discontinuous maps which are related to polygonal billiards and quantum chaos.

IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, 2006

This paper deals with the design and realization of bioinspired kinematic chains controllable bot... more This paper deals with the design and realization of bioinspired kinematic chains controllable both in position and compliance (or stiffness) from a static and a dynamic point of view. While position control is clearly referred to common geometrical lagrangian coordinates for the considered system, in order to deal with the compliance of the chain, especially in dynamic cases, global and

We investigate quantitative recurrence in systems having an infinite measure. We extend the Ornst... more We investigate quantitative recurrence in systems having an infinite measure. We extend the Ornstein-Weiss theorem for a general class of infinite systems estimating return time in decreasing sequences of cylinders. Then we restrict to a class of one dimensional maps with indifferent fixed points and calculate quantitative recurrence in sequences of balls, obtaining that this is related to the behavior of the map near the fixed points.

ABSTRACT We show an elementary method to have (finite time and asymptotic) computer assisted expl... more ABSTRACT We show an elementary method to have (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rate for systems satisfying a Lasota Yorke inequality. The bounds are deduced by the ones of suitable approximations of the system&#39;s transfer operator. We also present some rigorous experiment showing the approach and some concrete result.

Waiting time indicators are defined by measuring the time needed for a point x to approach a give... more Waiting time indicators are defined by measuring the time needed for a point x to approach a given point y. Our aim is to test the use of waiting time indicators, and quantitative recurrence ones to numerically estimate the local dimension of attractors in dynamical systems, as suggested by some rigorous results.

We survey an area of recent development, relating dynamics to theoretical computer science. We di... more We survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long term behavior of dynamical systems. We will focus on the statistical limiting behavior and invariant measures. We present a general method allowing the algorithmic approximation at any given accuracy of invariant measures. The method can be applied in many interesting cases, as we shall explain. On the other hand, we exhibit some examples where the algorithmic approximation of invariant measures is not possible. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem).

ABSTRACT Measuring and monitoring through wearable technology parameters related to human movemen... more ABSTRACT Measuring and monitoring through wearable technology parameters related to human movement, posture, and gesture are gaining momentum because of their wide range of potential applications in daily-life conditions. In previous studies, carbon elastomers (CEs) have been used as strain sensors. Recent developments of CE sensors mathematical modeling demonstrated that the CEs can be used as electrogoniometers. It was proved that for small local curvatures of CE layers, the resistance of a strip constituting a layer depends only on the total curvature of the same layer and not on the particular shape that the sensor keeps in adherence with a surface. Further, it was proved, theoretically and experimentally, that a double-layer configuration provides better accuracy with respect to a single-layer configuration. These results have been obtained under the hypothesis that the device was bent, but not extended. In this paper, we substituted the inextensible insulating layer in the sensors with an elastic one, allowing the system to extend its length. This improvement required further study to make it fit for biomechanical applications following epithelial deformations produced by joint movements and minimizes skin motion artifacts.

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point whi... more A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior as a typical point of the system.

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recu... more We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension.