3.
Douglas D. Novaes; Tere Seara; Marco A. Teixeira; Iris O. Zeli
Study of periodic orbits in periodic perturbations of planar reversible Filippov systems having a two-fold cycle Journal Article
Em: SIAM J. Appl. Dyn. Syst., vol. 19, não 2, pp. 1343-1371, 2020.
@article{NovSeaTeiZel2020,
title = {Study of periodic orbits in periodic perturbations of planar reversible Filippov systems having a two-fold cycle},
author = {Douglas D. Novaes and Tere Seara and Marco A. Teixeira and Iris O. Zeli },
url = {http://arxiv.org/abs/1910.01954},
doi = {10.1137/19M1289959},
year = {2020},
date = {2020-06-01},
urldate = {2020-06-01},
journal = {SIAM J. Appl. Dyn. Syst.},
volume = {19},
number = {2},
pages = {1343-1371},
abstract = {We study the existence of periodic solutions in a class of nonsmooth differential systems obtained from nonautonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed problem has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the existence of periodic solutions in a class of nonsmooth differential systems obtained from nonautonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a simple two-fold cycle, which is characterized by a closed trajectory connecting a visible two-fold singularity to itself. It is shown that under certain generic conditions the perturbed problem has sliding and crossing periodic solutions. In order to get our results, Melnikov's ideas were applied together with tools from the geometric singular perturbation theory. Finally, a study of a perturbed piecewise Hamiltonian model is performed.
2.
Jaume Llibre; Douglas D. Novaes; Iris O. Zeli
Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems Journal Article
Em: Revista de Matemátca Iberoamericana, vol. 36, pp. 291-318, 2020.
@article{LliNovZel2019,
title = {Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems},
author = {Jaume Llibre and Douglas D. Novaes and Iris O. Zeli},
url = {hyyp://dx.doi.org/10.4171/rmi/1131
https://arxiv.org/abs/1801.01730},
doi = {10.4171/rmi/1131},
year = {2020},
date = {2020-01-01},
urldate = {2020-01-01},
journal = {Revista de Matemátca Iberoamericana},
volume = {36},
pages = {291-318},
abstract = {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 $CZsubsetR^n$ of periodic solutions satisfying $dim(mathcal{Z})
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+ e F_1^n(x)+e^2F_2^n(x),
]
in $R^{d+2}$ where $e$ 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.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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 $CZsubsetR^n$ 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+ e F_1^n(x)+e^2F_2^n(x),
]
in $R^{d+2}$ where $e$ 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.
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+ e F_1^n(x)+e^2F_2^n(x),
]
in $R^{d+2}$ where $e$ 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.
1.
Douglas D. Novaes; Marco A. Teixeira; Iris O. Zeli
The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems Journal Article
Em: Nonlinearity, vol. 31, pp. 2083–2104, 2018.
@article{NovTeiZel2018,
title = {The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems},
author = {Douglas D. Novaes and Marco A. Teixeira and Iris O. Zeli},
url = {https://doi.org/10.1088/1361-6544/aaaaf7
https://arxiv.org/abs/1809.03433},
doi = {https://doi.org/10.1088/1361-6544/aaaaf7},
year = {2018},
date = {2018-04-06},
urldate = {2018-04-06},
journal = {Nonlinearity},
volume = {31},
pages = {2083–2104},
abstract = {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 of planar Filippov systems assuming that it presents a codimension-two minimal set for the critical value of the parameter. 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.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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 of planar Filippov systems assuming that it presents a codimension-two minimal set for the critical value of the parameter. 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.
