arXiv subjects

Luan V. M. F. Silva

Publications and source records attributed to Luan V. M. F. Silva.

5 recordsLinked to original sources

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.

math.DS↗

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.

math.DS↗

arXiv2024

A Melnikov analysis on a family of second order discontinuous differential equations

This paper aims to provide a Melnikov-like function that governs the existence of periodic solutions bifurcating from period annuli in certain families of second-order discontinuous differential equations of the form $\ddot{x}+α\; \textrm{sign}(x)=ηx+\varepsilon \;f(t,x,\dot{x})$. This family has attracted considerable attention from researchers, particularly in the analysis of specific instances of $f(t,x,\dot{x})$. The study of this type of differential equation is motivated by its significance in modeling systems with abrupt state changes, both in natural and engineering contexts.

math.DS↗

arXiv2024

On the boundedness of solutions of a forced discontinuous oscillator

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.

math.DS↗

arXiv2024

Invariant tori and boundedness of solutions of non-smooth oscillators with Lebesgue integrable forcing term

Since Littlewood works in the 1960's, the boundedness of solutions of Duffing-type equations $\ddot{x}+g(x)=p(t)$ has been extensively investigated. More recently, some researches have focused on the family of non-smooth forced oscillators $ \ddot{x}+\text{sgn}(x)=p(t)$, mainly because it represents a simple limit scenario of Duffing-type equations for when $g$ is bounded. Here, we provide a simple proof for the boundedness of solutions of the non-smooth forced oscillator in the case that the forcing term $p(t)$ is a $T$-periodic Lebesgue integrable function with vanishing average. We reach this result by constructing a sequence of invariant tori whose union of their interiors covers all the $(t,x,\dot x)$-space, $(t,x,\dot{x})\in \mathbb{S}^1\times\mathbb{R}^2$.

math.DS↗