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Marco Sabatini

Publications and source records attributed to Marco Sabatini.

13 recordsLinked to original sources
arXiv2023

Global injectivity of planar non-singular maps polynomial in one variable

We consider non-singular and Jacobian maps whose components are polynomial in the variable y. We prove that if a map has y-degree one, then it is the composition of a triangular map and a quasi-triangular map. We also prove that non-singular y-quadratic maps are injective if one of the leading functional coefficients does not vanish. Moreover, y-quadratic Jacobian maps are the composition of a quasi-triangular map and 3 triangular maps. Other results are given for wider classes of non-singular maps, considering also injectivity on vertical strips I x R.

math.DS↗
arXiv2022

Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space

Fessler and Gutierrez \cite{Fe,Gu} proved that if a non-singular planar map has Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective. We prove that the same holds replacing $(0,+\infty)$ with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove that a Jacobian map $(P,Q)$ is injective if $P_x + Q_y$ is not a surjective function.

math.CA↗
arXiv2018

Measure-preserving symmetries and reversibilities of ordinary differential systems

We prove that measure-preserving symmetries of an $n$-dimensional differential system preserve its divergence and the divergence derivatives along the solutions. Also, we prove that measure-preserving reversibilities preserve odd-order divergence derivatives along the solutions, and that even-order derivatives are multiplied by $-1$. We apply such results to find all the area-preserving symmetries and reversibilities of planar Lotka-Volterra and Liénard systems.

math.DS↗
arXiv2015

Every period annulus is both reversible and symmetric

We prove that for every planar differential system with a period annulus there exists an involution $σ$ such that the system is $σ$-symmetric. We also prove that for for every planar differential system with a period annulus there exist infinitely many involutions $σ$ such that the system is $σ$-reversible.

math.CA↗
arXiv2015

Centers with equal period functions

We give a sufficient condition for systems with symmetries to have periodic solutions with equal periods. We show that the main result can be applied both to Hamiltonian and to non-Hamiltonian systems. We apply the main results to produce planar centers with equal period functions.

math.CA↗
arXiv2012

The period functions' higher order derivatives

We prove a formula for the $n$-th derivative of the period function $T$ in a period annulus of a planar differential system. For $n = 1$, we obtain Freire, Gasull and Guillamon formula for the period's first derivative \cite{FGG}. We apply such a result to hamiltonian systems with separable variables and other systems. We give some sufficient conditions for the period function of conservative second order O.D.E.'s to be convex.

math.CA↗
arXiv2010

Existence and uniqueness of limit cycles in a class of second order ODE's with inseparable mixed terms

We prove a uniqueness result for limit cycles of the second order ODE $\ddot x + \dot x ϕ(x,\dot x) + g(x) = 0$. Under mild additional conditions, we show that such a limit cycle attracts every non-constant solution. As a special case, we prove limit cycle's uniqueness for an ODE studied in \cite{ETA} as a model of pedestrians' walk. This paper is an extension to equations with a non-linear $g(x)$ of the results presented in \cite{S}.

math.DS↗
arXiv2009

A note on the divergence-free Jacobian Conjecture in R^2

We give a shorter proof to a recent result by Neuberger, in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results presented in Neuberger.

math.AG↗