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Luciano Lopez

Università degli Studi di Bari, Mathematics, Faculty Member

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Papers by Luciano Lopez

Symplectic Mimetic Finite Difference methods for Hamiltonian wave equations in 2D

In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial ... more In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in time to integrate the semi-discrete Hamiltonian system. The main characteristic of MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach, associated with a symplectic method for the time integration yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper.

Special issue: Important aspects on structural dynamical systems and their numerical computations. Selected papers based on the presentations at the SDS08 meeting, Capitolo-Monopoli, Bari, Italy, June 17–20, 2008

Mathematics and Computers in Simulation, 2011

Special Issue Editorial: Discontinuous dynamical systems: Theory and numerical methods

A model for the hepatic glucose metabolism based on Hill and step functions

Journal of Computational and Applied Mathematics, 2015

We propose a system of first-order ordinary differential equations to describe and understand the... more We propose a system of first-order ordinary differential equations to describe and understand the physiological mechanisms of the interplay between plasma glucose and insulin and their behaviors in diabetes. The proposed model is based on Hill and step functions which are used to simulate the switch-like behavior that occurs in metabolic regulatory variables when some of the threshold parameters are approached. A simplified piecewise-linear system is also proposed the study the possible equilibria and solutions and used to introduce simple theoretical control mechanisms representing the action of an artificial pancreas and regulating exogenous insulin.

Spectral properties and conservation laws in Mimetic Finite Difference methods for PDEs

Journal of Computational and Applied Mathematics, 2015

Discontinuous dynamical systems: Theory and numerical methods

Mathematics and Computers in Simulation, 2014

Structural dynamical systems in linear algebra and control: computational aspects

Future Generation Computer Systems, 2003

Special section: Numerical methods for structured systems

Future Generation Computer Systems, 2006

UNIQUENESS OF FILIPPOV SLIDING VECTOR FIELD ON THE INTERSECTION OF TWO SURFACES IN R3 AND IMPLICATIONS FOR STABILITY OF PERIODIC ORBITS

Journal of Nonlinear Science, 2015

In this paper, we consider the class of smooth sliding Filippov vector fields in R3 on the inter... more In this paper, we consider the class of smooth sliding Filippov vector fields in
R3 on the intersection of two smooth surfaces: = 1 \2, where i = {x : hi(x) =
0}, and hi : R3 ! R, i = 1, 2, are smooth functions with linearly independent normals.
Although, in general, there is no unique Filippov sliding vector field on , here we prove
that –under natural conditions– all Filippov sliding vector fields are orbitally equivalent
on . In other words, the aforementioned ambiguity has no meaningful impact. We also
examine the implication of this result in the important case of a periodic orbit a portion
of which slides on .

One-Sided Event Location Techniques in the Numerical Solution of Discontinuous Differential Systems

BIT Numerical Mathematics, Dec 15, 2014

In this short paper, event location techniques for a differential system the solution of which i... more In this short paper, event location techniques for a differential system the
solution of which is directed towards a surface S defined as the 0-set of a smooth
function h: S = {x 2 Rn : h(x) = 0 } are considered. It is assumed that the exact
solution trajectory hits S non-tangentially, and numerical techniques guaranteeing
that the trajectory approaches S from one side only (i.e., does not cross it) are studied.
Methods based on Runge Kutta schemes which arrive to S in a finite number of
steps are proposed. The main motivation of this paper comes from integration of
discontinuous differential systems of Filippov type, where location of events is of
paramount importance.

ON THE EQUIVALENCE BETWEEN THE SIGMOIDAL APPROACH AND UTKIN’S APPROACH FOR PIECEWISE-LINEAR MODELS OF GENE REGULATORY NETWORKS

SIAM Journal on Applied Dynamical Systems , Jan 2015

This paper is concerned with piecewise linear dynamical systems modeling a simple class of gene r... more This paper is concerned with piecewise linear dynamical systems modeling a simple class of gene regulatory networks. One of the main issues when dealing with these problems, is that the vector field is not defined on the discontinuity hyperplanes. Two different methods are usually employed in literature to overcome this issue: Filippov's convexification approach and the steep sigmoidal approach. A particular selection of Filippov's vector field, namely Utkin's vector field, will be of interest to us. Our purpose is twofold: show that Utkin's vector field is well defined on the intersection Σ of two discontinuity hyperplanes (under assumptions of attractivity) and prove that, for Σ nodally attractive and attractive with three surfaces, Utkin's approach and the steep sigmoidal approach are equivalent, i.e., the corresponding solutions on Σ are the same. This allows to study the piecewise dynamical system, and hence the gene regulatory network it models, with no ambiguity.

Sharp sufficient attractivity conditions for sliding on a codimension 2 discontinuity surface

Mathematics and Computers in Simulation , Dec 2014

We consider Filippov sliding motion on a co-dimension 2 discontinuity surface. We give conditions... more We consider Filippov sliding motion on a co-dimension 2 discontinuity surface. We give conditions under which Σ is attractive through sliding which are sharper than those given in a previous paper of ours. Under these sharper conditions, we show that the sliding vector field considered in the same paper is still uniquely defined and varies smoothly in x ∈ Σ. A numerical example illustrate our results.

Numerical solution of discontinuous differential systems: approaching the discontinuity from one side,

Applied Numerical Mathematics

A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis

We consider sliding motion, in the sense of Filippov, on a discontinuity surface Σ of co-dimensio... more We consider sliding motion, in the sense of Filippov, on a discontinuity surface Σ of co-dimension 2. We characterize, and restrict to, the case of Σ being attractive through sliding. In this situation, we show that a certain Filippov sliding vector field fF (suggested in Alexander and Seidman, 1998 [2], di Bernardo et al., 2008 [6], Dieci and Lopez, 2011 [10]) exists and is unique.

On the continuous extension of Adams-Bashforth methods and the event location in discontinuous ODEs

The interpolation polynomial in the k-step Adams–Bashforth method may be used to compute the nume... more The interpolation polynomial in the k-step Adams–Bashforth method may be used to compute the numerical solution at off grid points. We show that such a numerical solution is equivalent to the one obtained by the Nordsieck technique for changing the step size. We also provide an application of this technique to the event location in discontinuous differential systems.

Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field

Abstract In this paper we consider the issue of sliding motion in Filippov systems on the interse... more Abstract In this paper we consider the issue of sliding motion in Filippov systems on the intersection of two or more surfaces. To this end, we propose an extension of the Filippov sliding vector field on manifolds of co-dimension p, with p≥ 2. Our model passes through the use of a multivalued sign function reformulation. To justify our proposal, we will restrict to cases where the sliding manifold is attractive.

A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side

This work is dedicated to the memory of Donato Trigiante who has been the first teacher of Numeri... more This work is dedicated to the memory of Donato Trigiante who has been the first teacher of Numerical Analysis of the second author. The authors remember Donato as a generous teacher, always ready to discuss with his students, able to give them profound and interesting suggestions. Here, we present a survey of numerical methods for differential systems with discontinuous right hand side.

Numerical Methods for Discontinuous Singularly Perturbed Differential Systems

Fundamental matrix solutions of piecewise smooth differential systems

ABSTRACT: We consider the fundamental matrix solution associated to piecewise smooth differential... more ABSTRACT: We consider the fundamental matrix solution associated to piecewise smooth differential systems of Filippov type, in which the vector field varies discontinuously as solution trajectories reach one or more surfaces. We review the cases of transversal intersection and of sliding motion on one surface. We also consider the case when sliding motion takes place on the intersection of two or more surfaces. Numerical results are also given.

Sliding motion in Filippov differential systems: Theoretical results and a computational approach

In this work, we discuss some theoretical and numerical aspects of solving differential equations... more