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Douglas Duarte Novaes

Publications and source records attributed to Douglas Duarte Novaes.

15 recordsLinked to original sources
arXiv2026

Maxwell Strata in the sub-Riemannian problem on solvable, nonnilpotent regular three-dimensional Lie groups

In this paper, we study the sub-Riemannian problem associated with contact structures on connected, simply connected, solvable, non-nilpotent, regular three-dimensional Lie groups. For these groups, the vertical component of the Hamiltonian system takes the form of a perturbed pendulum. A qualitative phase-space analysis allows us to prove that this vertical component exhibits nontrivial symmetries. In particular, we are able to fully characterize the Maxwell set corresponding to these symmetries, and show that its first Maxwell time coincides with the period of the pendulum for almost all geodesics. This result yields an explicit upper bound for the cut time in terms of the period of the pendulum.

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arXiv2021

A note on invariant measures for Filippov systems

We are interested in Filippov systems which preserve a probability measure on a compact manifold. We define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a consequence, the volume preserving Filippov systems are the refractive piecewise volume preserving ones. We conjecture that if a Filippov system admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. We prove this conjecture for Lipschitz differential inclusions. Then, in light of our previous results, we analyze the existence of invariant measures for many examples of Filippov systems defined on compact manifolds.

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arXiv2020

Sliding Shilnikov Connection in Filippov-type Predator-Prey Model

Recently, a piecewise smooth differential system was derived as a model of a 1 predator-2 prey interaction where the predator feeds adaptively on its preferred prey and an alternative prey. In such a model, strong evidence of chaotic behavior was numerically found. Here, we revisit this model and prove the existence of a Shilnikov sliding connection when the parameters are taken in a codimension one submanifold of the parameter space. As a consequence of this connection, we conclude, analytically, that the model behaves chaotically for an open region of the parameter space.

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arXiv2019

Asymptotic behavior of periodic solutions in one-parameter families of Liénard equations

In this paper, we consider one--parameter ($λ>0$) families of Liénard differential equations. We are concerned with the study on the asymptotic behavior of periodic solutions for small and large values of $λ>0$. To prove our main result we use the relaxation oscillation theory and a topological version of the averaging theory. More specifically, the first one is appropriate for studying the periodic solutions for large values of $λ$ and the second one for small values of $λ$. In particular, our hypotheses allow us to establish a link between these two theories.

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arXiv2019

Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems

The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non-autonomous $n$-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold $\mathcal{Z}\subset\mathbb{R}^n$ of periodic solutions satisfying $\dim(\mathcal{Z})<n.$ Then, we apply this result to study limit cycles bifurcating from periodic solutions of linear differential systems, $x'=Mx$, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the following differential system $x'=Mx+ \varepsilon F_1^n(x)+\varepsilon^2F_2^n(x),$ in $\mathbb{R}^{d+2}$ where $\varepsilon$ is a small parameter, $M$ is a $(d+2)\times(d+2)$ matrix having one pair of pure imaginary conjugate eigenvalues, $m$ zeros eigenvalues, and $d-m$ non-zero real eigenvalues.

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arXiv2017

The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems

Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for $k$-parameter families of planar vector fields. In the present study we focus on a qualitative analysis of $2$-parameter families, $Z_{α,β}$, of planar Filippov systems assuming that $Z_{0,0}$ presents a codimension-two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the first return map is nonvanishing. We analyzed the codimension-two scenario through the exhibition of its bifurcation diagram.

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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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arXiv2016

Chaos induced by sliding phenomena in Filippov systems

In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit. More specifically we prove that the first return map, defined nearby this orbit, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each natural number m it has infinitely many periodic points with period m.

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arXiv2015

On the periodic solutions of discontinuous piecewise differential systems

Motivated by problems coming from different areas of the applied science we study the periodic solutions of the following differential system $$x'(t)=F_0(t,x)+\varepsilon F_1(t,x)+\varepsilon^2 R(t,x,\varepsilon),$$ when $F_0$, $F_1$, and $R$ are discontinuous piecewise functions, and $\varepsilon$ is a small parameter. It is assumed that the manifold $\mathbb{Z}$ of all periodic solutions of the unperturbed system $x'=F_0(t,x)$ has dimension $n$ or smaller then $n$. The averaging theory is one of the best tools to attack this problem. This theory is completely developed when $F_0$, $F_1$ and $R$ are continuous functions, and also when $F_0=0$ for a class of discontinuous differential systems. Nevertheless does not exist the averaging theory for studying the periodic solutions of discontinuous differential system when $F_0\neq0$. In this paper we develop this theory for a big class of discontinuous differential systems.

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arXiv2014

A simple solution to the Braga-Mello conjecture

Recently Braga and Mello conjectured that for a given natural number n there is a piecewise linear system with two zones in the plane with exactly n limit cycles. In this paper we prove a result from which the conjecture is an immediate consequence. Several explicit examples are given where location and stability of limit cycles are provided.

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arXiv2014

On the birth of limit cycles for non-smooth dynamical systems

The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail.

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arXiv2012

Perturbed damped pendulum: finding periodic solutions

Using the damped pendulum system we introduce the averaging method to study the periodic solutions of a dynamical system with small perturbation. We provide sufficient conditions for the existence of periodic solutions with small amplitude of the non--linear perturbed damped pendulum. The averaging theory provides a useful means to study dynamical systems, accessible to Master and PhD students.

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arXiv2012

Physical assets replacement: an analytical approach

The economic life of an asset is the optimum length of its usefulness, which is the moment that the asset's expenses are minimum. In this paper, the economic life of physical assets, such as industry machine and equipment, can be interpreted as the moment that the minimum is reached by its equivalent property cost function, defined as the sum of all equivalent capital and maintenance costs during its life. Many authors in classical papers have used principles of engineering economic to solve the assets replacement problem. However, in the literature, the main attributes found were proved with intuitive ideas instead mathematical analysis. Therefore, in this paper the main goal is to study these principles of engineering economic with mathematical techniques. Here, is used non-smooth analysis to classify all the possibilities for the minimum of a class of equivalent property cost functions of assets. The minimum of these function gives the optimum moment for the asset to be replaced, i.e., its economic life.

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