2.
Tiago de Carvalho; Douglas D. Novaes; Durval J. Tonon
Sliding motion on tangential sets of Filippov systems Journal Article
Em: Preprint, 2021.
@article{CarNovTon2021,
title = {Sliding motion on tangential sets of Filippov systems},
author = {Tiago de Carvalho and Douglas D. Novaes and Durval J. Tonon},
url = {https://arxiv.org/abs/2111.12377},
year = {2021},
date = {2021-11-25},
journal = {Preprint},
abstract = {We consider piecewise smooth vector fields $Z=(Z_+, Z_-)$ defined in $R^n$ where both vector fields are tangent to the switching manifold $Sigma$ along a manifold $M$. Our main purpose is to study the existence of an invariant vector field defined on $M$, that we call {it tangential sliding vector field}. We provide necessary and sufficient conditions under $Z$ to characterize the existence of this vector field. Considering a regularization process, we proved that this tangential sliding vector field is topological equivalent to an invariant dynamics under the slow manifold. Besides then, at the end of the work we apply the results in a system that governs the intermittent treatment of HIV.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We consider piecewise smooth vector fields $Z=(Z_+, Z_-)$ defined in $R^n$ where both vector fields are tangent to the switching manifold $Sigma$ along a manifold $M$. Our main purpose is to study the existence of an invariant vector field defined on $M$, that we call {it tangential sliding vector field}. We provide necessary and sufficient conditions under $Z$ to characterize the existence of this vector field. Considering a regularization process, we proved that this tangential sliding vector field is topological equivalent to an invariant dynamics under the slow manifold. Besides then, at the end of the work we apply the results in a system that governs the intermittent treatment of HIV.
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.