2.
Kamila S. Andrade; Otávio M.L. Gomide; Douglas D. Novaes; Marco A. Teixeira
Bifurcation Diagrams of Global Connections in Filippov Systems Journal Article
Em: Nonlinear Analysis: Hybrid Systems, vol. 50, pp. 101397, 2023.
@article{AndGomNov2023,
title = {Bifurcation Diagrams of Global Connections in Filippov Systems},
author = {Kamila S. Andrade and Otávio M.L. Gomide and Douglas D. Novaes and Marco A. Teixeira },
url = {http://arxiv.org/abs/1905.11950},
doi = {10.1016/j.nahs.2023.101397},
year = {2023},
date = {2023-11-01},
urldate = {2023-11-01},
journal = {Nonlinear Analysis: Hybrid Systems},
volume = {50},
pages = {101397},
abstract = {In this paper, we are concerned about the qualitative behaviour of planar Filippov systems around some typical minimal sets, namely, polycycles. In the smooth context, a polycycle is a simple closed curve composed by a collection of singularities and regular orbits, inducing a first return map. Here, this concept is extended to Filippov systems by allowing typical singularities lying on the switching manifold. Our main goal consists in developing a method to investigate the unfolding of polycycles in Filippov systems. In addition, we applied this method to describe bifurcation diagrams of Filippov systems around certain polycycles. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper, we are concerned about the qualitative behaviour of planar Filippov systems around some typical minimal sets, namely, polycycles. In the smooth context, a polycycle is a simple closed curve composed by a collection of singularities and regular orbits, inducing a first return map. Here, this concept is extended to Filippov systems by allowing typical singularities lying on the switching manifold. Our main goal consists in developing a method to investigate the unfolding of polycycles in Filippov systems. In addition, we applied this method to describe bifurcation diagrams of Filippov systems around certain polycycles.
1.
Kamila S. Andrade; Oscar A. R. Cespedes; Dayane Cruz; Douglas D. Novaes
Higher order Melnikov analysis of planar piecewise linear vector fields with nonlinear switching curve Journal Article
Em: Journal of Differential Equations, vol. 287, pp. 1-36, 2021.
@article{AndCesCruNov2021,
title = {Higher order Melnikov analysis of planar piecewise linear vector fields with nonlinear switching curve},
author = {Kamila S. Andrade and Oscar A. R. Cespedes and Dayane Cruz and Douglas D. Novaes},
url = {https://arxiv.org/abs/2006.11352},
doi = {10.1016/j.jde.2021.03.039},
year = {2021},
date = {2021-03-29},
urldate = {2021-03-29},
journal = {Journal of Differential Equations},
volume = {287},
pages = {1-36},
abstract = {In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)geq 4,$ $H(3)geq 8,$ $H(n)geq7,$ for $ngeq 4$ even, and $H(n)geq9,$ for $ngeq 5$ odd. This improves all the previous results for $ngeq2.$ Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov Theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)geq 4,$ $H(3)geq 8,$ $H(n)geq7,$ for $ngeq 4$ even, and $H(n)geq9,$ for $ngeq 5$ odd. This improves all the previous results for $ngeq2.$ Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov Theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.
