arXiv subjects

Otavio Henrique Perez

Publications and source records attributed to Otavio Henrique Perez.

11 recordsLinked to original sources

arXiv2026

Cyclicity of sliding cycles in regularizations of piecewise linear two-folds

We study limit cycles produced by Sotomayor-Teixeira regularizations of piecewise linear vector fields with generic two-fold singularities. We focus on the visible-visible and visible-invisible cases where sliding cycles occur, treating cycles from both sides of the switching manifold in a unified way. In further details, by relating the cyclicity of sliding cycles and zeros of the slow divergence integral, we prove that the cyclicity of compact families of sliding cycles is bounded by two when such integral does not vanish identically. In contrast to previous works, we do not perform a case-by-base study, but instead relate zeros of slow divergence integrals to crossing limit cycles of a suitably defined auxiliary piecewise linear system. For visible folds, we provide necessary and sufficient conditions that assure the existence of a unique simple zero of the slow divergence integral, which implies the existence of two limit cycles. We also show that, when a hyperbolic singularity of the sliding vector field lies at the boundary of the sliding segment, then the cyclicity is bounded by one.

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arXiv2026

Generalized upper principal part of real planar polynomial vector fields

The goal of this paper is to generalize recent results about the topological classification of the dynamics of a real planar polynomial vector field near infinity. Given a polynomial vector field $X$, using Newton polyhedra one can define its generalized upper principal part $X_Γ^{U}$. By dropping some monomials of $X_Γ^{U}$, we define its minimal generalized upper principal part $X_{G}^{U}$. We prove that there exist an open and dense set $\widetilde{\mathfrak{U}}_{1}$ in the set of polynomial vector fields with Newton degenerate upper principal part satisfying the following property: if $X\in\widetilde{\mathfrak{U}}_{1}$, then $X$ and $X_{G}^{U}$ are topologically equivalent near infinity, provided that $X_{G}^{U}$ satisfies some additional non-degeneracy assumptions. Our techniques rely on toric compactification and the Normal Form Theorem.

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arXiv2026

An unbounded number of canard limit cycles in linear regularizations of piecewise linear systems

The purpose of this paper is to study the number of limit cycles of canard type in linear regularizations of piecewise linear systems with non-monotonic transition functions. Using the notion of slow divergence integral and elementary breaking mechanisms, we construct systems with an arbitrary finite number of hyperbolic limit cycles. The Hopf breaking mechanism deals with transition functions with precisely one critical point in the interval $(-1,1)$. On the other hand, the jump breaking mechanism produces any number of limit cycles using transition functions with precisely three critical points in $(-1,1)$.

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arXiv2025

Canard cycles of non-linearly regularized piecewise smooth vector fields

The main purpose of this paper is to study limit cycles in non-linear regularizations of planar piecewise smooth systems with fold points (or more degenerate tangency points) and crossing regions. We deal with a slow fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for upper bounds and the existence of limit cycles of canard type, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of piecewise linear center such that for any integer $k>0$ it has at least $k+1$ limit cycles, for a suitably chosen monotonic transition function $φ_k:\mathbb{R}\rightarrow\mathbb{R}$. We prove a similar result for regularized invisible-invisible fold-fold singularities of type II$_2$. Canard cycles of dodging layer are also considered, and we prove that such limit cycles undergo a saddle-node bifurcation.

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arXiv2025

Cyclicity of sliding cycles with singularities of regularized piecewise smooth visible-invisible two-folds

In this paper we study the cyclicity of sliding cycles for regularized piecewise smooth visible-invisible two-folds, in the presence of singularities of the Filippov sliding vector field located away from two-folds. We obtain a slow-fast system after cylindrical blow-up and use a well-known connection between the divergence integral along orbits and transition maps for vector fields. Since properties of the divergence integral depend on the location and multiplicity of singularities, we divide the sliding cycles into different classes, which can then produce different types of cyclicity results. As an example, we apply our results to regularized piecewise linear systems.

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arXiv2024

Fractal analysis of canard cycles and slow-fast Hopf points in piecewise smooth Liénard equations

The main goal of this paper is to give a complete fractal analysis of piecewise smooth (PWS) slow-fast Liénard equations. For the analysis, we use the notion of Minkowski dimension of one-dimensional orbits generated by slow relation functions. More precisely, we find all possible values for the Minkowski dimension near PWS slow-fast Hopf points and near bounded balanced crossing canard cycles. We study fractal properties of the unbounded canard cycles using PWS classical Liénard equations. We also show how the trivial Minkowski dimension implies the non-existence of limit cycles of crossing type close to Hopf points. This is not true for crossing limit cycles produced by bounded balanced canard cycles, i.e. we find a system undergoing a saddle-node bifurcation of crossing limit cycles and a system without limit cycles (in both cases, the Minkowski dimension is trivial). We also connect the Minkowski dimension with upper bounds for the number of limit cycles produced by bounded canard cycles.

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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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arXiv2023

Topological equivalence in the infinity of a planar vector field and its principal part defined through Newton polytope

Given a planar polynomial vector field $X$ with a fixed Newton polytope $\mathcal{P}$, we prove (under some non degeneracy conditions) that the monomials associated to the upper boundary of $\mathcal{P}$ determine (under topological equivalence) the phase portrait of $X$ in a neighbourhood of boundary of the Poincaré--Lyapunov disk. This result can be seen as a version of the well known result of Berezovskaya, Brunella and Miari for the dynamics at the infinity, We also discuss the effect of the Poincaré--Lyapunov compactification on the Newton polytope.

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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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