arXiv subjects

Paulo Ricardo da Silva

Publications and source records attributed to Paulo Ricardo da Silva.

13 recordsLinked to original sources

arXiv2026

Bifurcation of limit cycles in piecewise holomorphic systems separated by two external circles

This work investigates the number of limit cycles bifurcating from the periodic orbits of a linear center when it is perturbed by piecewise holomorphic systems with three zones separated by two external circles. We analyze how different geometric configurations of these zones influence the number of limit cycles. Moreover, we show the existence of at least sixteen limit cycles in fourth-degree piecewise polynomial holomorphic systems under certain configurations.

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arXiv2025

On the Piecewise Holomorphic Systems with Three Zones

We study piecewise-smooth systems with three zones, $\dot{z} = f_i(z)$, $i = 1,2,3,$ whose discontinuity set $Σ$ consists either of a pair of parallel lines or a pair of circles tangent to each other internally or externally. Each $f_i:\overline{\mathbb{C}} \to \overline{\mathbb{C}}$ is assumed to be a holomorphic function. We establish conditions ensuring the existence of limit cycles in such systems and provide lower bounds for the maximum number of limit cycle. Our approach combines the Melnikov method, local integrability properties of holomorphic systems, and the existence of normal forms around zeros and poles.

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arXiv2024

Polynomial slow-fast systems on the Poincaré-Lyapunov sphere

The main goal of this paper is to study compactifications of polynomial slow-fast systems. More precisely, the aim is to give conditions in order to guarantee normal hyperbolicity at infinity of the Poincaré-Lyapunov sphere for slow-fast systems defined in $\mathbb{R}^{n}$. For the planar case, we prove a global version of the Fenichel Theorem, which assures the persistence of invariant manifolds in the whole Poincaré-Lyapunov disk. We also discuss the appearence of non normally hyperbolic points at infinity, namely: fold, transcritical and pitchfork singularities.

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arXiv2022

Slow-fast normal forms arising from piecewise smooth vector fields

We studied piecewise smooth differential systems of the form $$\dot{z} = Z(z) = \dfrac{1 + \operatorname{sgn}(F)}{2}X(z) + \dfrac{1 - \operatorname{sgn}(F)}{2}Y(z),$$ where $F: \mathbb{R}^{n}\rightarrow \mathbb{R}$ is a smooth map having 0 as a regular value. We consider linear regularizations of the vector field $Z$ given by $$\dot{z}= Z_{\varepsilon}(z) = \dfrac{1 + φ(F/\varepsilon)}{2}X(z) +\dfrac{1 - φ(F /\varepsilon)}{2}Y(z),$$where $φ$ is a transition function (not necessarily monotonic) and nonlinear regularizations of the vector field $Z$ whose transition function is monotonic. It is a well-known fact that the regularized system is a slow-fast system. The main contribution of this paper is the study of typical singularities of slow-fast systems that arise from (linear or nonlinear) regularizations. We developed an algorithm to construct suitable transition functions, and we apply these ideas in order to create slow-fast singularities from normal forms of piecewise smooth vector fields. We present examples of transition functions that, after regularization of a PSVF normal form, generate normally hyperbolic, fold, transcritical, and pitchfork singularities.

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arXiv2021

On the classification of planar constrained differential systems under topological equivalence

This paper concerns the local study of analytic constrained differential systems (or impasse systems) of the form $A(x)\dot{x} = F(x)$, $x\in\mathbb{R}^{2}$, where $F$ is a vector field and $A$ is a matrix valued function. Using techniques of resolution of singularities with weighted blow ups, we extend well-known results in the literature on topological classification of phase portraits of planar vector fields to the context of such class of systems. We also study and classify phase portraits of constrained systems near singular points of the so called impasse set $Δ=\{x:\det A (x) = 0\}$.

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arXiv2020

Resolution of singularities of 2-dimensional real analytic constrained differential systems

We present a theorem of resolution of singularities for real analytic constrained differential systems $A(x)\dot{x} = F(x)$ defined on a 2-manifold with corners having impasse set $\{x; \det A(x) = 0\}$. This result can be seen as a generalization of the classical one for 2-dimensional real analytic vector fields. Our proof is based on weighted blow-ups and the Newton polygon.

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arXiv2020

Phase Portrait of the Riccati Quadratic Polynomial Differential Systems

In this paper we characterize the phase portrait of the Riccati quadratic polynomial differential systems $$\dot{x}= α_2(x),\quad\dot{y} = ky^2+β_1(x) y + γ_2(x), $$ with $(x,y)\in\mathbb{R}^2$, $γ_2(x)$ non-zero (otherwise the system is a Bernoulli differential system), $k\neq0$ (otherwise the system is a Lienard differential system), $β_ 1(x)$ a polynomial of degree at most $1$, $α_ 2(x)$ and $γ_ 2(x)$ polynomials of degree at most 2, and the maximum of the degrees of $ α_2(x)$ and $k y^2+β_1(x) y + γ_2(x)$ is 2. We give the complete description of their phase portraits in the Poincare disk

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arXiv2020

Invariant Algebraic Surfaces and Constrained Systems

We study flows of smooth vector fields $X$ over invariant surfaces $M$ which are levels of rational first integrals. It leads us to study constrained systems, that is, systems with impasses. We identify a subset $\mathcal{I} \subset M$ which we call "pseudo-impasse" set and analyze the flow of X by points of $\mathcal{I}$. Systems well known in the literature exemplify our results: Lorenz, Chen, Falkner-Skan and Fisher-Kolmogorov. We also study 1-parameter families of integrable systems and unfolding of minimal sets. Our main tool is the geometric singular perturbation theory.

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arXiv2018

On non-smooth slow-fast systems

We deal with non-smooth differential systems $\dot{z}=X(z), z\in R^{n},$ with discontinuity occurring in a codimension one smooth surface $Σ$. A regularization of $X$ is a 1-parameter family of smooth vector fields $X^δ,δ>0$, satisfying that $X^δ$ converges pointwise to $X$ in $R^{n}\setminusΣ$, when $δ\rightarrow 0$. We work with two known regularizations: the classical one proposed by Sotomayor and Teixeira and its generalization, using non-monotonic transition functions. Using the techniques of geometric singular perturbation theory we study minimal sets of regularized systems. Moreover, non-smooth slow-fast systems are studied and the persistence of the sliding region by singular perturbations is analyzed.

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arXiv2018

Slow--fast systems and sliding on codimension 2 switching manifolds

In this work we consider piecewise smooth vector fields $X$ defined in $\R^n\setminus Σ$, where $Σ$ is a self-intersecting switching manifold. A double regularization of $X$ is a 2-parameter family of smooth vector fields $X_{\e.η}$, $\e,η>0,$ satisfying that $X_{\e,η}$ converges pointwise to $X$ on $\R^n\setminusΣ$, when $\e,η\rightarrow 0$. We define the sliding region on the non regular part of $Σ$ as a limit of invariant manifolds of $X_{\e.η}$. Since the double regularization provides a slow--fast system, the GSP-theory (geometric singular perturbation theory) is our main tool.

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arXiv2017

Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

We consider piecewise smooth vector fields (PSVF) defined in open sets $M\subseteq R^n$ with switching manifold being a smooth surface $Σ$. The PSVF are given by pairs $X = (X_+, X_-)$, with $X = X_+$ in $Σ_+$ and $X = X_-$ in $Σ_-$ where $Σ_+$ and $Σ_-$ are the regions on $M$ separated by $Σ.$ A regularization of $X$ is a 1-parameter family of smooth vector fields $X^ε,ε>0,$ satisfying that $X^ε$ converges pointwise to $X$ on $M\setminusΣ$, when $ε\rightarrow 0$. Inspired by the Fenichel Theory , the sliding and sewing dynamics on the discontinuity locus $Σ$ can be defined as some sort of limit of the dynamics of a nearby smooth regularization $X^ε$. While the linear regularization requires that for every $ε>0$ the regularized field $X^ε$ is in the convex combination of $X_+ $ and $X_- $ the nonlinear regularization requires only that $X^ε$ is in a continuous combination of $X_+ $ and $X_- $. We prove that for both cases, the sliding dynamics on $Σ$ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. \end{abstract}

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arXiv2017

Regularization of Discontinuous Foliations: Blowing up and Sliding Conditions via Fenichel Theory

We study the regularization of an oriented 1-foliation $\mathcal{F}$ on $M \setminus Σ$ where $M$ is a smooth manifold and $Σ\subset M$ is a closed subset, which can be interpreted as the discontinuity locus of $\mathcal{F}$. In the spirit of Filippov's work, we define a sliding and sewing dynamics on the discontinuity locus $Σ$ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions.

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arXiv2010

Canard Cycles and Poincaré Index of Non-Smooth Vector Fields on the Plane

This paper is concerned with closed orbits of non-smooth vector fields on the plane. For a subclass of non-smooth vector fields we provide necessary and sufficient conditions for the existence of canard kind solutions. By means of a regularization we prove that the canard cycles are singular orbits of singular perturbation problems which are limit periodic sets of a sequence of limit cycles. Moreover, we generalize the Poincaré Index for non-smooth vector fields.

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