1.
João L. Cardoso; Jaume Llibre; Douglas D. Novaes; Durval J. Tonon
Simultaneous occurrence of sliding and crossing limit cycles in Filippov systems Journal Article
Em: Dynamical Systems: An International Journal, vol. 35, não 3, pp. 490-514, 2020.
@article{CarLliNovTon2020,
title = {Simultaneous occurrence of sliding and crossing limit cycles in Filippov systems},
author = {João L. Cardoso and Jaume Llibre and Douglas D. Novaes and Durval J. Tonon},
url = {https://arxiv.org/abs/1905.06427},
doi = {10.1080/14689367.2020.1722064},
year = {2020},
date = {2020-01-29},
journal = {Dynamical Systems: An International Journal},
volume = {35},
number = {3},
pages = {490-514},
abstract = {In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line x=0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a
real one for y<0 and a virtual one for y>0, and such that the real center is a global center. Then working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one more crossing limit cycle can appear. In addition, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov functions, the Extended Chebyshev systems and the Bendixson
transformation.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line x=0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a
real one for y<0 and a virtual one for y>0, and such that the real center is a global center. Then working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one more crossing limit cycle can appear. In addition, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov functions, the Extended Chebyshev systems and the Bendixson
transformation.
real one for y<0 and a virtual one for y>0, and such that the real center is a global center. Then working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one more crossing limit cycle can appear. In addition, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov functions, the Extended Chebyshev systems and the Bendixson
transformation.
