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Marco Spadini

Publications and source records attributed to Marco Spadini.

16 recordsLinked to original sources
arXiv2025

Forced oscillations for generalized $\Phi$-Laplacian equations with Carath\'eodory perturbations

Using topological methods, we study the structure of the set of forced oscillations of a class of parametric, implicit ordinary differential equations with a generalized $\Phi$-Laplacian type term. We work in the Carath\'eodory setting. Under suitable assumptions, involving merely the Brouwer degree in Euclidean spaces, we obtain global bifurcation results. In some illustrative examples we provide a visual representation of the bifurcating set.

math.CA↗
arXiv2023

An introduction to topological degree in Euclidean spaces

This paper aims to provide a careful and self-contained introduction to the theory of topological degree in Euclidean spaces. It is intended for people mostly interested in analysis and, in general, a heavy background in algebraic or differential topology is not required.

math.FA↗
arXiv2022

Periodic perturbations of a class of scalar second order functional differential equations

We study, by means of a topological approach, the forced oscillations of second order functional retarded differential equations subject to periodic perturbations. We consider a delay-type functional dependence involving a gamma probability distribution. By a linear chain trick we obtain a first order system of ODE's whose $T$-periodic solutions correspond to those of the functional equation.

math.CA↗
arXiv2018

Surface shear waves in a half-plane with depth-variant structure

We consider the propagation of surface shear waves in a half-plane, whose shear modulus $μ(y)$ and density $ρ(y)$ depend continuously on the depth coordinate $y$. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency $ω$ and wave number $k$ as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum $ω(k)$ for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of $N(k)$ surface wave solutions, with $N(k) \to \infty$ as $k \to \infty$. The most intriguing result is a possibility of the existence of infinite number of solutions, $N(k)=\infty$, for any given $k$. These three options are conditioned by the properties of $μ(y)$ and $ρ(y)$.

math.CA↗
arXiv2014

Periodic perturbations with delay of coupled differential equations on manifolds with application to a sunflower-like equation

We investigate the structure of the set of $T$-periodic solutions to periodically perturbed coupled delay differential equations on differentiable manifolds. By using fixed point index and degree-theoretic methods we prove the existence of branches of $T$-periodic solutions to the considered equations. As main application of our methods, we study a generalized version of the sunflower equation.

math.CA↗
arXiv2014

About the notion of non-$T$-resonance and applications to topological multiplicity results for ODEs on differentiable manifolds

By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for $T$-periodic solutions of parametrized $T$-periodic perturbations of autonomous ODEs on a differentiable manifold $M$. In order to provide insights into the key notion of $T$-resonance, we consider the elementary situations $M = \mathbb{R}$ and $M = \mathbb{R}^2$. So doing, we provide more comprehensive analysis of those cases and find improved conditions.

math.CA↗
arXiv2014

Periodic perturbations of constrained motion problems on a class of implicitly defined manifolds

We study forced oscillations on differentiable manifolds which are globally defined as the zero set of appropriate smooth maps in some Euclidean spaces. Given a T-periodic perturbative forcing field, we consider the two different scenarios of a nontrivial unperturbed force field and of perturbation of the zero field. We provide simple, degree-theoretic conditions for the existence of branches of T-periodic solutions. We apply our construction to a class of second order Differential-Algebraic Equations.

math.CA↗
arXiv2013

Periodic solutions of semi-explicit differential-algebraic equations with time-dependent constraints

In this paper we investigate the properties of the set of T-periodic solutions of semi-explicit parametrized Differential-Algebraic Equations with non-autonomous constraints of a particular type. We provide simple, degree theoretic conditions for the existence of branches of T-periodic solutions of the considered equations. Our approach is based on topological arguments about differential equations on implicitly defined manifolds, combined with elementary facts of matrix analysis.

math.CA↗
arXiv2012

On a class of differential-algebraic equations with infinite delay

We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed Differential-Algebraic Equations, allowing the perturbation to contain a distributed and possibly infinite delay. Under suitable assumptions, the perturbed equations are equivalent to Retarded Functional (Ordinary) Differential Equations on a manifold. Our study is based on known results about the latter class of equations.

math.CA↗
arXiv2012

Branches of forced oscillations for a class of constrained ODEs: a topological approach

We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a $k$-dimensional differentiable manifold $M \subseteq \mathbb{R}^m$. We assume that $M$ is globally defined as the zero set of a smooth map and, as a first step, we determine a formula which reduces the computation of the degree of a tangent vector field on $M$ to the Brouwer degree of a suitable map in $\mathbb{R}^m$. As further applications, we study the set of harmonic solutions to periodic semi-esplicit differential-algebraic equations.

math.CA↗
arXiv2011

Local inversion of planar maps with nice nondifferentiability structure

When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and common vertex at the origin) our main result provides a sufficient condition for any map $L$, that is continuous and piecewise linear relatively to this slicing, to be invertible. Some examples show that the assumptions of the theorem cannot be relaxed too much. In particular, convexity of the slices cannot be dropped altogether when $n=4$. This result cannot be plainly extended to a greater number of slices. Our result is proved by a combination of linear algebra and topological arguments.

math.CA↗
arXiv2010

Bang--bang trajectories with a double switching time: sufficient strong local optimality conditions

This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of the ones considered in previous papers for more specific cases. We require both the strict bang-bang Legendre condition, and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory.

math.OC↗
arXiv2009

A note on topological methods for a class of Differential-Algebraic Equations

We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the computation of the degree of the associated tangent vector field that does not require any explicit knowledge of the manifold. We use this formula to study the set of harmonic solutions to periodic perturbations of our equations. Two different classes of applications are provided.

math.CA↗
arXiv2008

Periodic perturbations with delay of autonomous differential equations on manifolds

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré $T$-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.

math.DS↗