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Alessandro Calamai

Publications and source records attributed to Alessandro Calamai.

At least 19 recordsLinked to original sources
arXiv2026

An affine Birkhoff-Kellogg type result in product spaces and its application to differential systems

We prove a version of the Birkhoff-Kellogg theorem in product spaces, under affine transformations. This is a fairly natural framework that arises when dealing with parameter-dependent systems of functional-differential equations. We provide criteria on the existence of parameter-dependent solutions that have all the components nontrivial. The theoretical results are illustrated in the case of systems of second order functional-differential equations, where the functional part can cover the interesting cases of time and state-dependent deviated arguments. We provide a concrete example, for which we furnish numerically approximated solutions, coherent with our theoretical framework, and in which we compute or estimate all the constants that are required by our abstract results.

math.DS↗
arXiv2025

Birkhoff-Kellogg type results in product spaces and their application to differential systems

We provide a new version of the well-known Birkhoff-Kellogg invariant-direction Theorem in product spaces. Our results concern operator systems and give the existence of component-wise eigenvalues, instead of scalar eigenvalues as in the classical case, that have corresponding eigenvectors with all components nontrivial and localized by their norm. We also show that, when applied to nonlinear eigenvalue problems for differential equations, this localization property of the eigenvectors provides, in turn, qualitative properties of the solutions. This is illustrated in two contexts of systems of PDEs and ODEs. We show the applicability of our theoretical results with two explicit examples.

math.FA↗
arXiv2025

A new construction for Melnikov chaos in piecewise-smooth planar systems

In this paper we consider a piecewise smooth $2$-dimensional system \[ \dot{\vec{x}}=\vec{g} (\vec{x})+\varepsilon\vec{g}(t,\vec{x},\varepsilon) \] where $\varepsilon>0$ is a small parameter and $\vec{f}$ is discontinuous along a curve $\Omega^0$. We assume that $\vec{0}$ is a critical point for any $\varepsilon \geq 0$, and that for $\varepsilon=0$ the system admits a trajectory $\vec{\gamma}(t)$ homoclinic to $\vec{0}$ and crossing transversely $\Omega^0$ in $\vec{\gamma}(0)$. In a previous paper we have shown that, also in an $n$-dimensional setting, the classical Melnikov condition is enough to guarantee the persistence of the homoclinic to perturbations, but more recently we have found an open condition, a geometric obstruction which is not possible in the smooth case, which prevents chaos for $2$-dimensional systems when $\vec{g}$ is periodic in $t$. In this paper we show that when this obstruction is removed we have chaos as in the smooth case. The proofs involve a new construction of the set $\Sigma$ from which the chaotic pattern originates. The results are illustrated by examples.

math.DS↗
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↗
arXiv2025

Some contributions on Melnikov chaos for smooth and piecewise-smooth planar systems: "trajectories chaotic in the future"

We consider a $2$-dimensional autonomous system subject to a $1$-periodic perturbation, i.e. $$ \dot{\vec{x}}=\vec{f}(\vec{x})+\epsilon\vec{g}(t,\vec{x},\epsilon),\quad \vec{x}\in\Omega .$$ We assume that for $\epsilon=0$ there is a trajectory $\vec{\gamma}(t)$ homoclinic to the origin which is a critical point: in this context Melnikov theory provides a sufficient condition for the insurgence of a chaotic pattern when $\epsilon \ne 0$. In this paper we show that for any line $\Xi$ transversal to $\{\vec{\gamma}(t) \mid t \in \mathbb{R} \}$ and any $\tau \in [0,1]$ we can find a set $\Sigma^+(\Xi,\tau)$ of initial conditions giving rise to a pattern chaotic just in the future, located in $\Xi$ at $t=\tau$. Further diam$(\Sigma^+(\Xi,\tau)) \le \epsilon^{(1+\nu)/ \underline{\sigma}}$ where $\underline{\sigma}>0$ is a constant and $\nu>0$ is a parameter that can be chosen as large as we wish. The same result holds true for the set $\Sigma^-(\Xi,\tau)$ of initial conditions giving rise to a pattern chaotic just in the past. In fact all the results are developed in a piecewise-smooth context, assuming that $\vec{0}$ lies on the discontinuity curve $\Omega^0$: we recall that in this setting chaos is not possible if we have sliding phenomena close to the origin. This paper can also be considered as the first part of the project to show the existence of classical chaotic phenomena when sliding close to the origin is not present.

math.DS↗
arXiv2024

On the dynamics of non-autonomous systems in a~neighborhood of a~homoclinic trajectory

This article is devoted to the study of a $2$-dimensional piecewise smooth (but possibly) discontinuous dynamical system, subject to a non-autonomous perturbation; we assume that the unperturbed system admits a homoclinic trajectory $\vec{\gamma}(t)$. Our aim is to analyze the dynamics in a neighborhood of $\vec{\gamma}(t)$ as the perturbation is turned on, by defining a Poincar\'e map and evaluating fly time and space displacement of trajectories performing a loop close to $\vec{\gamma}(t)$. Besides their intrinsic mathematical interest, these results can be thought of as a first step in the analysis of several interesting problems, such as the stability of a homoclinic trajectory of a non-autonomous ODE and a possible extension of Melnikov chaos to a discontinuous setting.

math.DS↗
arXiv2024

On the solvability of parameter-dependent elliptic functional BVPs on annular-like domains

We investigate the existence of nontrivial solutions of parameter-dependent elliptic equations with deviated argument in annular-like domains in $\mathbb{R}^{n}$, with $n\geq 2$, subject to functional boundary conditions. In particular we consider a boundary value problem that may be used to model heat-flow problems. We obtain an existence result by means of topological methods; in particular, we make use of a recent variant in affine cones of the celebrated Birkhoff--Kellogg theorem. Using an ODE argument, we illustrate in an example the applicability of our theoretical result.

math.AP↗
arXiv2024

A Birkhoff-Kellogg type theorem for discontinuous operators with applications

By means of fixed point index theory for multi-valued maps, we provide an analogue of the classical Birkhoff--Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary differential equations with discontinuities. We illustrate in details this fact for a class of second order boundary value problem with deviated arguments and discontinuous terms. In a specific example, we explicitly compute the terms that occur in our theory.

math.CA↗
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↗
arXiv2021

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.

math.SP↗
arXiv2021

A degree associated to linear eigenvalue problems in Hilbert spaces and applications to nonlinear spectral theory

We extend to the infinite dimensional context the link between two completely different topics recently highlighted by the authors: the classical eigenvalue problem for real square matrices and the Brouwer degree for maps between oriented finite dimensional real manifolds. Thanks to this extension, we solve a conjecture regarding global continuation in nonlinear spectral theory that we have formulated in a recent article. Our result (the ex conjecture) is applied to prove a Rabinowitz type global continuation property of the solutions to a perturbed motion equation containing an air resistance frictional force.

math.SP↗
arXiv2020

Nonzero positive solutions of fractional Laplacian systems with functional terms

We study the existence of non-zero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with non-local functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach.

math.AP↗
arXiv2020

Boundary value problems associated with singular strongly nonlinear equations with functional terms

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type $$\big(Φ(k(t)\,x'(t))\big)' + f(t,\mathcal{G}_x(t))\,ρ(t, x'(t)) = 0$$ on a compact interval $[a,b]$. These equations are quite general due to the presence of a strictly increasing homeomorphism $Φ$, the so-called $Φ$-Laplacian operator, of a nonnegative function $k$, which may vanish on a set of null measure, and moreover of a functional term $\mathcal{G}_x$. We look for solutions, in a suitable weak sense, which belong to the Sobolev space $W^{1,1}([a,b])$. Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.

math.CA↗
arXiv2019

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces

We consider the nonlinear eigenvalue problem $Lx + \varepsilon N(x) = λCx$, $\|x\|=1$, where $\varepsilon,λ$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and $N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $Σ$ of the solutions $(x,\varepsilon,λ) \in S \times \mathbb R\times \mathbb R$ of this problem. Namely, if the operators $N$ and $C$ are compact, under suitable assumptions on a solution $p_*=(x_*,0,λ_*)$ of the unperturbed problem, we prove that the connected component of $Σ$ containing $p_*$ is either unbounded or meets a triple $p^*=(x^*,0,λ^*)$ with $p^* \not= p_*$. When $C$ is the identity and $G=H$ is finite dimensional, the assumptions on $(x_*,0,λ_*)$ mean that $x_*$ is an eigenvector of $L$ whose corresponding eigenvalue $λ_*$ is simple. Therefore, we extend a previous result obtained by the authors in the finite dimensional setting. Our work is inspired by a paper of R. Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space.

math.SP↗
arXiv2019

The Brouwer degree associated to classical eigenvalue problems and applications to nonlinear spectral theory

Thanks to a connection between two completely different topics, the classical eigenvalue problem in a finite dimensional real vector space and the Brouwer degree for maps between oriented differentiable real manifolds, we were able to solve, at least in the finite dimensional context, a conjecture regarding global continuation in nonlinear spectral theory that we formulated in some recent papers. The infinite dimensional case seems nontrivial, and is still unsolved.

math.SP↗