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result(s) for
"Duarte Novaes, Douglas"
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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.
Journal Article
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.
Paper
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.
Paper
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})
Paper
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.
Paper
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 \\(\\Sigma\\). The PSVF are given by pairs \\(X = (X_+, X_-)\\), with \\(X = X_+\\) in \\(\\Sigma_+\\) and \\(X = X_-\\) in \\(\\Sigma_-\\) where \\(\\Sigma _+\\) and \\(\\Sigma _-\\) are the regions on \\(M\\) separated by \\(\\Sigma.\\) A regularization of \\(X\\) is a 1-parameter family of smooth vector fields \\(X^{\\epsilon},\\epsilon>0,\\) satisfying that \\(X^{\\epsilon}\\) converges pointwise to \\(X\\) on \\(M\\setminus\\Sigma\\), when \\(\\epsilon\\rightarrow 0\\). Inspired by the Fenichel Theory , the sliding and sewing dynamics on the discontinuity locus \\(\\Sigma\\) can be defined as some sort of limit of the dynamics of a nearby smooth regularization \\(X^{\\epsilon}\\). While the linear regularization requires that for every \\(\\epsilon>0\\) the regularized field \\(X^{\\epsilon}\\) is in the convex combination of \\(X_+ \\) and \\(X_- \\) the nonlinear regularization requires only that \\(X^{\\epsilon}\\) is in a continuous combination of \\(X_+ \\) and \\(X_- \\). We prove that for both cases, the sliding dynamics on \\(\\Sigma\\) is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. \\end{abstract}
Paper
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_{\\alpha,\\beta}\\), 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.
Paper
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.
Paper
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.
Paper
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.
Paper