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Paulo Santana

Publications and source records attributed to Paulo Santana.

At least 19 recordsLinked to original sources
arXiv2026

Hopf-like bifurcations induced by hysteresis and time-delay near monodromic tangential singularities

We investigate planar piecewise-analytic vector fields, focusing on the merged focus and other monodromic tangential singularities when hysteresis or time-delay is incorporated into the switching condition. If the singularity is an asymptotically stable solution of the system with an instantaneous switch, then the introduction of hysteresis or time-delay causes an attracting limit cycle to be formed locally. We derive asymptotic expressions for the size and period of the limit cycle, allowing any degrees of tangency and any order for the smallest non-zero Lyapunov coefficient. We find that the growth rate of the limit cycle differs for hysteresis and time-delay, and differs to that of the related pseudo-Hopf bifurcation.

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arXiv2026

Homoclinic Theorems for piecewise smooth vector fields

In this paper we provide extensions of the Birkhoff-Smale Homoclinic Theorem and the $λ$-Lemma for piecewise smooth vector fields free of sliding motion. We prove that if the vector field is $T$-periodic in its independent variable and has a transverse homoclinic point, then its time-$T$-map is conjugated with a Bernoulli Shift and thus the vector field is chaotic. The Bernoulli Shifts considered here may have any finite number of symbols, or countably infinite many.

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arXiv2026

Piecewise continuous and monotonic maps on the interval

Let $f$ be a piecewise continuous and monotonic map on the interval with at most finitely many discontinuities and turning points. In this paper we study properties about this class of maps and show its main difference from the continuous case. We define and study the notion of closed structure, which can be seen as an generalization of periodic orbit. We also study the periodic orbits that are away from the discontinuities of $f$, extending the notion of trapped and free orbits.

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arXiv2025

Limit cycles and invariant algebraic curves

We give lower bounds in terms of~$n,$ for the number of limit cycles of polynomial vector fields of degree~$n,$ having any prescribed invariant algebraic curve. By applying them when the ovals of this curve are also algebraic limit cycles we obtain a new recurrent property for the Hilbert numbers. Finally, we apply our results to two important families of models: Kolmogorov systems and a general family of systems appearing in Game Theory.

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arXiv2025

Planar vector fields without invariant algebraic curves

In this work we revisit and extend the method introduced by Lins Neto, Sad and Scárdua for detecting the non-existence of invariant algebraic curves other than some prescribed invariant nodal curve. We prove that, under the existence of a suitable example, the space of polynomial vector fields whose elements have the prescribed curve as their unique invariant algebraic curve is residual and of full measure. We apply this framework to Kolmogorov vector fields, showing that generically the coordinate axes are the unique invariant algebraic curves. Finally, we also refine existing characterizations related to Hilbert's 16th problem, showing that if there exists a bound for the number of limit cycles of a vector field of degree n, then it can be attained by a vector field without algebraic limit cycles.

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arXiv2025

The hybrid matching of Hurwitz systems

In this paper we study planar hybrid systems composed by two stable linear systems, defined by Hurwitz matrices, in addition with a jump that can be a piecewise linear, a polynomial or an analytic function. We provide an explicit analytic necessary and sufficient condition for this class of hybrid systems to be asymptotically stable. We also prove the existence of limit cycles in this class of hybrid systems. Our results can be seen as generalizations of results already obtained in the literature. This was possible due to an embedding of piecewise smooth vector fields in a hybrid structure.

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arXiv2025

Generalized Pseudo-Hopf Bifurcation: Limit Cycle Position and Period

We investigate planar piecewise-smooth vector fields with a discontinuity line, focusing on the bifurcation of crossing limit cycles that arise when one of the vector fields is translated along the discontinuity set. We establish topological conditions under which such bifurcations occur and, under additional generic hypotheses, derive precise asymptotic expressions for both the position and the period of the resulting limit cycle in terms of the perturbation parameter. Our results extend the classical pseudo-Hopf bifurcation: they are not restricted to invisible folds or elementary monodromic singularities, but also apply to nilpotent centers/foci, half-monodromic singularities such as cusps, periodic orbits, and hyperbolic polycycles, thereby encompassing both local and non-local configurations. We show that the period function exhibits distinct asymptotic behaviors depending on the interacting objects. In particular, we provide a comprehensive table summarizing the leading terms of the period and position of the limit cycle for each possible configuration.

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arXiv2025

A Piecewise Smooth $λ$-Lemma

In this paper we provide extensions of the $λ$-Lemma (also known as Inclination Lemma) for piecewise smooth vector fields and maps. In order to achieve our main result, we investigate the regularity of time-T-maps of piecewise smooth vector fields defined at crossing orbits. We prove that these maps are homeomorphisms and also piecewise smooth diffeomorphisms.

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arXiv2025

On the cyclicity of persistent hyperbolic polycycles

In this work we consider families of smooth vector fields having a persistent polycycle with $n$ hyperbolic saddles. We derive the asymptotic expansion of the return map associated to the polycycle, determining explicitly its leading terms. As a consequence, explicit conditions on the leading terms allow us to determine the cyclicity of such polycycles. We then apply our results to study the cyclicity of a polycycle of a model with applications in Game Theory.

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arXiv2025

On the cyclicity of hyperbolic polycycles

Let $X$ be a planar smooth vector field with a polycycle $Γ^n$ with $n$ sides and all its corners, that are at most $n$ singularities, being hyperbolic saddles. In this paper we study the cyclicity of $Γ^n$ in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least $k,$ for any $k\leqslant n.$ Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when $X$ is polynomial there is a polynomial perturbation (in general with degree much higher that the one of $X$) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world.

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arXiv2024

On the stability of hybrid polycycles

In this paper we provide the stability of generic polycycles of hybrid planar vector fields, extending previous known results in the literature. The polycycles considered here may have hyperbolic saddles, tangential singularities and jump singularities.

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arXiv2024

On the limit cycles of a quartic model for evolutionary stable strategies

This paper studies the number of centers and limit cycles of the family of planar quartic polynomial vector fields that has the invariant algebraic curve $(4x^2-1)(4y^2-1)=0.$ The main interest for this type of vector fields comes from their appearance in some mathematical models in Game Theory composed by two players. In particular, we find examples with five nested limit cycles surrounding the same singularity, as well as examples with four limit cycles formed by two disjoint nests, each one of them with two limit cycles. We also prove a Berlinski\u ı's type result for this family of vector fields.

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arXiv2024

Limit cycles and chaos in planar hybrid systems

In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic dynamics.

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arXiv2024

A note on Hilbert 16th Problem

Let $\mathcal{H}(n)$ be the maximum number of limit cycles that a planar polynomial vector field of degree $n$ can have. In this paper we prove that $\mathcal{H}(n)$ is realizable by structurally stable vector fields with only hyperbolic limit cycles and that it is a strictly increasing function whenever it is finite.

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arXiv2024

On a variant of Hilbert's 16th problem

We study the number of limit cycles that a planar polynomial vector field can have as a function of its number $m$ of monomials. We prove that the number of limit cycles increases at least quadratically with $m$ and we provide good lower bounds for $m\leqslant10$.

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arXiv2024

On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

It is known that non-hyperbolic limit cycles are structurally unstable in the set of planar smooth and analytical vector fields. In the polynomial case, it is known only that limit cycles of even degree are structurally unstable. In this paper, we prove that non-hyperbolic limit cycles of odd degree are also structurally unstable in the polynomial case, if we consider Whitney's topology.

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arXiv2023

On structural stability of Evolutionary Stables Strategies

The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar polynomial vector field with an invariant octothorpe. In this paper we provide sufficient and necessary conditions for structural stability in this model.

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