5.
Victoriano Carmona; Fernando Fernández Sánchez; Douglas D. Novaes
Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region Journal Article
Em: Communications in Nonlinear Science and Numerical Simulation, vol. 123, pp. 107257, 2023.
@article{CarFerNov2023b,
title = {Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region},
author = {Victoriano Carmona and Fernando Fernández Sánchez and Douglas D. Novaes},
url = {https://arxiv.org/abs/2207.14634},
doi = {10.1016/j.cnsns.2023.107257},
year = {2023},
date = {2023-08-01},
urldate = {2023-08-01},
journal = {Communications in Nonlinear Science and Numerical Simulation},
volume = {123},
pages = {107257},
abstract = {In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincaré half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincaré half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived.
4.
Victoriano Carmona; Fernando Fernández Sánchez; Douglas D. Novaes
A succinct characterization of period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness Journal Article
Em: Journal of Nonlinear Science, vol. 33, iss. 5, não 88, pp. 1-13, 2023.
@article{CarFerNov2022c,
title = {A succinct characterization of period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness},
author = {Victoriano Carmona and Fernando Fernández Sánchez and Douglas D. Novaes},
url = {http://arxiv.org/abs/2212.09063},
doi = {10.1007/s00332-023-09947-5},
year = {2023},
date = {2023-07-17},
urldate = {2023-07-17},
journal = {Journal of Nonlinear Science},
volume = {33},
number = {88},
issue = {5},
pages = {1-13},
abstract = {We close the problem of the existence of period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness. In fact, a characterization for the existence of such objects is provided by means of a few basic operations on the parameters.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We close the problem of the existence of period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness. In fact, a characterization for the existence of such objects is provided by means of a few basic operations on the parameters.
3.
Victoriano Carmona; Fernando Fernández Sánchez; Eli García-Medina; Douglas D. Novaes
Properties of Poincaré half-maps for planar linear systems and some direct applications to periodic orbits of piecewise systems Journal Article
Em: Electronic Journal of Qualitative Theory of Differential Equations, vol. 2023, iss. 22, pp. 1-18, 2023.
@article{CarFerNov203c,
title = {Properties of Poincaré half-maps for planar linear systems and some direct applications to periodic orbits of piecewise systems},
author = {Victoriano Carmona and Fernando Fernández Sánchez and Eli García-Medina and Douglas D. Novaes},
url = {https://arxiv.org/abs/2109.12673},
doi = {10.14232/ejqtde.2023.1.22},
year = {2023},
date = {2023-05-24},
urldate = {2023-04-21},
journal = {Electronic Journal of Qualitative Theory of Differential Equations},
volume = {2023},
issue = {22},
pages = {1-18},
abstract = {This paper deals with fundamental properties of Poincaré half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincaré half-maps, their series expansions (Taylor and Newton-Puiseux) at the tangency point and at infinity, the relative position between the graph of Poincaré half-maps and the bisector of the fourth quadrant, and the sign of their second derivatives. All these properties are essential to understand the dynamic behavior of planar piecewise linear systems. Accordingly, we also provide some of their most immediate, but non-trivial, consequences regarding periodic orbits.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
This paper deals with fundamental properties of Poincaré half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincaré half-maps, their series expansions (Taylor and Newton-Puiseux) at the tangency point and at infinity, the relative position between the graph of Poincaré half-maps and the bisector of the fourth quadrant, and the sign of their second derivatives. All these properties are essential to understand the dynamic behavior of planar piecewise linear systems. Accordingly, we also provide some of their most immediate, but non-trivial, consequences regarding periodic orbits.
2.
Victoriano Carmona; Fernando Fernández Sánchez; Douglas D. Novaes
Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line Journal Article
Em: Applied Mathematics Letters, vol. 137, pp. 108501, 2023.
@article{CarFerNov2022b,
title = {Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line},
author = {Victoriano Carmona and Fernando Fernández Sánchez and Douglas D. Novaes},
url = {https://arxiv.org/abs/2210.12125},
doi = {10.1016/j.aml.2022.108501},
year = {2023},
date = {2023-03-01},
journal = {Applied Mathematics Letters},
volume = {137},
pages = {108501},
abstract = {The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum-Chua's work, it has remained an open question whether this uniform upper bound exists or not. Here, we give a positive answer for this question by establishing the existence of a natural number $L^*leq 8$ for which any planar piecewise linear differential system with two zones separated by a straight line has no more than $L^*$ limit cycles. The proof is obtained by combining a newly developed integral characterization of Poincaré half-maps for linear differential systems with an extension of Khovanskii's theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum-Chua's work, it has remained an open question whether this uniform upper bound exists or not. Here, we give a positive answer for this question by establishing the existence of a natural number $L^*leq 8$ for which any planar piecewise linear differential system with two zones separated by a straight line has no more than $L^*$ limit cycles. The proof is obtained by combining a newly developed integral characterization of Poincaré half-maps for linear differential systems with an extension of Khovanskii's theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields.
1.
Victoriano Carmona; Fernando Fernández Sánchez; Douglas D. Novaes
A new simple proof for Lum-Chua's conjecture Journal Article
Em: Nonlinear Analysis: Hybrid Systems, vol. 40, pp. 100992-100999, 2021.
@article{CarFerNov2021,
title = {A new simple proof for Lum-Chua's conjecture},
author = {Victoriano Carmona and Fernando Fernández Sánchez and Douglas D. Novaes},
url = {http://arxiv.org/abs/1911.01372},
doi = {10.1016/j.nahs.2020.100992},
year = {2021},
date = {2021-05-01},
urldate = {2021-05-01},
journal = {Nonlinear Analysis: Hybrid Systems},
volume = {40},
pages = {100992-100999},
abstract = {The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel characterization for Poincaré half-maps in planar linear systems.
This proof is very short and straightforward, because this characterization avoids the inherent flaws of the usual methods to study piecewise linear systems (the appearance of large case-by-case analysis due to the unnecessary discrimination between the different spectra of the involved matrices, to deal with transcendental equations forced by the implicit occurrence of flight time, ...).
In addition, the application of the characterization allow us to prove that if a limit cycle exists, then it is hyperbolic and its stability is determined by a simple relationship between the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle and this simple expression for its stability have not been pointed out before.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel characterization for Poincaré half-maps in planar linear systems.
This proof is very short and straightforward, because this characterization avoids the inherent flaws of the usual methods to study piecewise linear systems (the appearance of large case-by-case analysis due to the unnecessary discrimination between the different spectra of the involved matrices, to deal with transcendental equations forced by the implicit occurrence of flight time, ...).
In addition, the application of the characterization allow us to prove that if a limit cycle exists, then it is hyperbolic and its stability is determined by a simple relationship between the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle and this simple expression for its stability have not been pointed out before.
This proof is very short and straightforward, because this characterization avoids the inherent flaws of the usual methods to study piecewise linear systems (the appearance of large case-by-case analysis due to the unnecessary discrimination between the different spectra of the involved matrices, to deal with transcendental equations forced by the implicit occurrence of flight time, ...).
In addition, the application of the characterization allow us to prove that if a limit cycle exists, then it is hyperbolic and its stability is determined by a simple relationship between the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle and this simple expression for its stability have not been pointed out before.
