Searcharxiv⌕ Search

arXiv subjects

Ricardo Miranda Martins

Publications and source records attributed to Ricardo Miranda Martins.

12 recordsLinked to original sources
arXiv2025

Limit cycles of piecewise smooth vector fields on torus with non-regular switching manifold

We study the maximum number of crossing limit cycles in piecewise-smooth vector fields on the two-dimensional torus, where the discontinuity set is the boundary of the fundamental square. Under the assumption of a polynomial first integral of degree n, we apply algebraic curve-intersection methods to obtain sharp upper bounds on cycle counts. In the quadratic case (n=2), we prove at most one cycle of type aa or bb and at most two of type aba or bab, and we give explicit parameter criteria for their existence. For general n, we show that cycles of type bb are bounded by n-1 and those of type aba by n(n-1). Finally, we construct concrete examples achieving these bounds, demonstrating their optimality.

math.DS↗
arXiv2024

On the existence of closed trajectories and pseudo-trajectories for a family of third order differential equations

The goal of this article is to study the existence of closed trajectories for the differential equation $\dddot{z}+a\ddot{z}+b\dot{z}+abz=\varepsilon F(z,\dot{z},\ddot{z})$ in two situations. In the first situation, we consider $F(z,\dot{z},\ddot{z})=1$ and $b={\rm sgn}(h(z,\dot{z},\ddot{z}))$, where $h(z,\dot{z},\ddot{z})=z^2+(\dot{z})^2+(\ddot{z})^2-1$. We show that the differential equation is equivalent to a piecewise smooth differential system that admits the unit sphere as the discontinuity manifold. We obtain conditions for the existence of a closed pseudo-trajectory in this case. In the second situation, we consider $\varepsilon \neq 0$ sufficiently small, $b>0$, and $F(z,\dot{z},\ddot{z})$ a $n$-degree polynomial. We show that the unperturbed differential equation has a family of isochronous periodic solutions filling an invariant plane. Then, we study the maximum number of limit cycles which bifurcate from this 2-dimensional isochronous using the averaging theory. Thus, within the same family, we have periodic solutions (in the case where the parameters create a smooth equation) and also pseudo-periodic solutions (in the case of Filippov systems).

math.DS↗
arXiv2023

Sharp estimates for the number of limit cycles in discontinuous generalized Liénard equations

In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations $\dot{x}=y, \ \dot{y}=-x-\varepsilon \cdot (f(x)\cdot y +{\rm sgn}(y)\cdot g(x))$. Using the averaging method, we were able to generalize a previous result for Liénard systems. In our generalization, we consider $g$ as a polynomial of degree $m$. We conclude that for sufficiently small values of $|ε|$, the number $\left[\frac{n}{2}\right]+\left[\frac{m}{2}\right]+1$ serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$. Furthermore, we demonstrate that it is indeed possible to achieve such a number of limit cycles.

math.DS↗
arXiv2022

Limit cycles for classes of piecewise smooth differential equations separated by the unit circle

In this article we study the existence of limit cycles in families of piecewise smooth differential equations having the unit circle as discontinuity region. We consider families presenting singularities of center or saddle type, visible or invisible, as well as the case without singularities. We establish an upper bound for the number of limit cycles and give examples showing that the maximum number of limit cycles can be reached. We also discuss the existence of homoclinic cycles for such differential equations in the saddle-center case.

math.DS↗
arXiv2021

Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto like theorems characterizing semi-local structural stability.

math.DS↗
arXiv2016

A numerical treatment to the problem of the quantity of Einstein metrics on flag manifolds

In this paper we employ numerical methods to study the Einstein equation \[ Ric(g)=λ\, g, \] where $Ric$ is the Ricci tensor and $λ$ is the Einstein constant, restricted to a class of full flag manifolds. These metrics describe the gravitational field of a vacuum with cosmological constant (vacuum is the case $λ=0$). In particular, we give estimates to the number of such metrics on the full flag manifolds $SU(n+1)/T^n$ for $n=4,5$, improving some classical estimatives. We also examine the isometric problem for these Einstein metrics. Our method can be applied for any fixed $n$.

math.DG↗
arXiv2016

Chaos in Piecewise Smooth Vector Fields on Two Dimensional Torus and Sphere

In this paper we study the global dynamics of piecewise smooth vector fields defined in the two dimensional torus and sphere. We provide conditions under these families exhibits periodic and dense trajectories and we describe some global bifurcations. We also study its minimal sets and characterize the chaotic behavior of the piecewise smooth vector fields defined in torus and sphere.

math.DS↗
arXiv2012

On non-smooth vector fields having a torus or a sphere as the sliding manifold

In this paper we consider a non-smooth vector field $Z=(X,Y)$, where $X,Y$ are linear vector fields in dimension 3 and the discontinuity manifold $Σ$ of $Z$ is or the usual embedded torus or the unitary sphere at origin. We suppose that $Σ$ is a sliding (stable/unstable) manifold with tangencies, by considering $X,Y$ inelastic over $Σ$. In each case, we study the tangencies of the vector field $Z$ with $Σ$ and describe the behavior of the trajectories of the sliding vector field over $Σ$: they are basically closed.

math.DS↗
arXiv2009

Global behavior of the Ricci flow on homogeneous manifolds with two isotropy summands

In this paper we study the global behavior of the Ricci flow equation for two classes of homogeneous manifolds with two isotropy summands. Using methods of the qualitative theory of differential equations, we present the global phase portrait of such systems and derive some geometrical consequences on the structure of such manifolds under the action of the Ricci flow.

math.DG↗
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.

math.DS↗
arXiv2009

The Ricci flow of left invariant metrics on full flag manifold SU(3)/T from a dynamical systems point of view

In this paper we study the behavior of the Ricci flow at infinity for the full flag manifold $SU(3)/T$ using techniques of the qualitative theory of differential equations, in special the Poincaré Compactification and Lyapunov exponents. We prove that there are four invariant lines for the Ricci flow equation, each one associated with a singularity corresponding to a Einstein metric. In such manifold, the bi-invariant normal metric is Einstein. Moreover, around each invariant line there is a cylinder of initial conditions such that the limit metric under the Ricci flow is the corresponding Einstein metric; in particular we obtain the convergence of left-invariant metrics to a bi-invariant metric under the Ricci flow.

math.DG↗