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Marco Antonio Teixeira

Publications and source records attributed to Marco Antonio Teixeira.

11 recordsLinked to original sources
arXiv2022

Foliations, solvability and global injectivity

Let $F: \mathbb{R}^n\to\mathbb{R}^n$ be a $C^{\infty}$ map such that $DF(x)$ is invertible for every $x\in\mathbb{R}^n$. Although being a local diffeomorphism, $F$ is not necessarily globally injective if $n\geq2$. Finding additional assumptions implying the global injectivity of $F$ for $n\geq 2$ is object of intense study in several areas of Mathematics. In this paper we revisit some assumptions and relations between them in the bidimensional case and discuss the natural higher dimensional situation.

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arXiv2017

The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems

Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for $k$-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of $2$-parameter families, $Z_{α,β}$, of planar Filippov systems assuming that $Z_{0,0}$ presents a codimension-two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the first return map is nonvanishing. We analyzed the codimension-two scenario through the exhibition of its bifurcation diagram.

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arXiv2015

Attractivity, degeneracy and codimension of a typical singularity in 3D piecewise smooth vector fields

We address the problem of understanding the dynamics around typical singular points of $3D$ piecewise smooth vector fields. A model $Z_0$ in $3D$ presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: \textit{(i)} $Z_0$ has an invariant plane $π_0$ filled up with periodic orbits (this means that the restriction $Z_0 |_{π_0}$ is a center around the singularity), \textit{(ii)} All trajectories of $Z_0$ converge to the surface $π_0$, and such attraction occurs in a very non-usual and amazing way, \textit{(iii)} given an arbitrary integer $k\geq 0$ then $Z_0$ can be approximated by $π_0$-invariant piecewise smooth vector fields $Z_{\varepsilon}$ such that the restriction $Z_{\varepsilon} |_{π_0}$ has exactly $k$-hyperbolic limit cycles, \textit{(iv)} the origin can be chosen as an asymptotic stable equilibrium of $Z_{\varepsilon}$ when $k=0$, and finally, \textit{(v)} $Z_0$ has infinite codimension in the set of all $3D$ piecewise smooth vector fields.

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arXiv2014

On the birth of limit cycles for non-smooth dynamical systems

The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail.

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arXiv2011

Two-fold symmetric singularity

We explore some qualitative dynamics in the neighborhood of the $3-dimensional$ two-fold symmetric singularity. We study the existence of an one-parameter family of regular (pseudo) periodic orbits of such systems near a reversible two-fold singularity.

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arXiv2011

On global linearization of planar involutions

Let $ϕ:\R^2\to\R^2$ be an orientation--preserving $C^1$ involution such that $ϕ(0)=0$ and let ${\rm Spc}\,(ϕ)=\{{\rm Eigenvalues\,\,of}\,\, Dϕ(p)\mid p\in\R^2\}$. We prove that if ${\rm Spc}\,{(ϕ)}\subset\R$ or ${\rm Spc}\,(ϕ)\cap [1,1+ε)=\emptyset$ for some $ε>0$ then $ϕ$ is globally $C^1$ conjugate to the linear involution $Dϕ(0)$ via the conjugacy $h=(I+Dϕ(0)ϕ)/2$, where $I:\R^2\to\R^2$ is the identity map. Similarly, if $ϕ$ is an orientation-reversing $C^1$ involution such that $ϕ(0)=0$ and ${\rm Trace}\,\big(Dϕ(0)Dϕ(p)\big)>-1 $ for all $p\in\R^2$ then $ϕ$ is globally $C^1$ conjugate to the linear involution $Dϕ(0)$ via the conjugacy $h$. Finally, we show that $h$ may fail to be a global linearization of $ϕ$ if the above conditions are not fulfilled.

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arXiv2010

Branching of periodic orbits in reversible hamiltonian systems

This paper deals with the dynamics of time-reversible Hamiltonian vector fields with 2 and 3 degrees of freedom around an elliptic equilibrium point in presence of symplectic involutions. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques used are Birkhoff and Belitskii normal forms combined with the Liapunov-Schmidt reduction.

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arXiv2009

Reversible-equivariant systems and matricial equations

This paper uses tools in group theory and symbolic computing to give a classification of the representations of finite groups with order lower than 9 that can be derived from the study of local reversible-equivariant vector fields in $\rn{4}$. The results are obtained by solving algebraically matricial equations. In particular, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitiskii normal form.

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