13.
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.
12.
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.
11.
Douglas D. Novaes; Marco A. Teixeira
Shilnikov problem in Filippov dynamical systems Journal Article
Em: Chaos, vol. 29, pp. 063110, 2019.
@article{NovTei2019,
title = {Shilnikov problem in Filippov dynamical systems},
author = {Douglas D. Novaes and Marco A. Teixeira},
url = {https://doi.org/10.1063/1.5093067
http://arxiv.org/abs/1504.02425},
doi = {10.1063/1.5093067},
year = {2019},
date = {2019-06-20},
journal = {Chaos},
volume = {29},
pages = {063110},
abstract = {In this paper we introduce the concept of sliding Shilnikov orbits for $3$D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A version of the Shilnikov's Theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first exhibiting such a sliding phenomenon.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we introduce the concept of sliding Shilnikov orbits for $3$D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A version of the Shilnikov's Theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first exhibiting such a sliding phenomenon.
10.
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.
9.
Paulo R. Guimarães; Douglas D. Novaes; Sérgio F. Reis; Marco A. Teixeira
The complexity-stability debate and May's paradox: A perspective from bifurcation theory Journal Article
Em: Forthcoming, 2016.
@article{GuiNovReiTei2016,
title = {The complexity-stability debate and May's paradox: A perspective from bifurcation theory},
author = {Paulo R. Guimarães and Douglas D. Novaes and Sérgio F. Reis and Marco A. Teixeira},
year = {2016},
date = {2016-01-01},
journal = {Forthcoming},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
8.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
Maximum number of limit cycles for certain piecewise linear dynamical systems Journal Article
Em: Nonlinear Dynamics, vol. 82, pp. 1159-1175, 2015.
@article{LliNovTeiND2015,
title = {Maximum number of limit cycles for certain piecewise linear dynamical systems},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://dx.doi.org/10.1007/s11071-015-2223-x},
doi = {10.1007/s11071-015-2223-x},
year = {2015},
date = {2015-07-23},
journal = {Nonlinear Dynamics},
volume = {82},
pages = {1159-1175},
abstract = {This paper deals with the question of the determinacy of the maximum number of limit cycles of some classes of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line Sigma. We restrict ourselves to the non-sliding limit cycles case, i.e. limit cycles that do not contain any sliding segment. Among all cases treated here, it is proved that the maximum number of limit cycles is at most 2 if one of the two linear differential systems of the discontinuous piecewise linear differential system has a focus in line of discontinuity, a center, or a weak saddle. We use the theory of Chebyshev systems for establishing sharp upper bounds for the number of limit cycles. Some normal forms are also provided for these systems.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
This paper deals with the question of the determinacy of the maximum number of limit cycles of some classes of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line Sigma. We restrict ourselves to the non-sliding limit cycles case, i.e. limit cycles that do not contain any sliding segment. Among all cases treated here, it is proved that the maximum number of limit cycles is at most 2 if one of the two linear differential systems of the discontinuous piecewise linear differential system has a focus in line of discontinuity, a center, or a weak saddle. We use the theory of Chebyshev systems for establishing sharp upper bounds for the number of limit cycles. Some normal forms are also provided for these systems.
7.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
On the periodic solutions of perturbed 4D non-resonant systems Journal Article
Em: The São Paulo Journal of Mathematical Sciences, vol. 9, não 2, pp. 229-250, 2015.
@article{LliNovTeiSPJMS2015,
title = {On the periodic solutions of perturbed 4D non-resonant systems},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://link.springer.com/article/10.1007/s40863-015-0017-1},
doi = {10.1007/s40863-015-0017-1},
year = {2015},
date = {2015-01-01},
journal = {The São Paulo Journal of Mathematical Sciences},
volume = {9},
number = {2},
pages = {229-250},
abstract = {We provide sufficient conditions for the existence of periodic solutions of a 4D non-resonant system perturbed by smooth or non-smooth functions. We apply these results to study the small amplitude periodic solutions of the non-linear planar double pendulum perturbed by smooth or non-smooth function.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We provide sufficient conditions for the existence of periodic solutions of a 4D non-resonant system perturbed by smooth or non-smooth functions. We apply these results to study the small amplitude periodic solutions of the non-linear planar double pendulum perturbed by smooth or non-smooth function.
6.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones Journal Article
Em: International Journal of Bifurcation and Chaos, vol. 25, não 11, pp. 1550144, 2015.
@article{LliNovTeiJBC2015,
title = {Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://dx.doi.org/10.1142/S0218127415501448},
doi = {10.1142/S0218127415501448},
year = {2015},
date = {2015-01-01},
journal = {International Journal of Bifurcation and Chaos},
volume = {25},
number = {11},
pages = {1550144},
abstract = {We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.
5.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
On the birth of limit cycles for non-smooth dynamical systems Journal Article
Em: Bulletin des Sciences Mathématiques, vol. 139, não 3, pp. 229 - 244, 2015.
@article{LliNovTeiBSM2015,
title = {On the birth of limit cycles for non-smooth dynamical systems},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://dx.doi.org/10.1016/j.bulsci.2014.08.011},
doi = {10.1016/j.bulsci.2014.08.011},
year = {2015},
date = {2015-01-01},
journal = {Bulletin des Sciences Mathématiques},
volume = {139},
number = {3},
pages = {229 - 244},
abstract = {The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non-smooth dynamical systems theory. An application is presented in careful detail.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non-smooth dynamical systems theory. An application is presented in careful detail.
4.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
Periodic solutions of Lienard differential equations via averaging theory of order two Journal Article
Em: Anais da Academia Brasileira de Ciências, vol. 87, não 4, pp. 1905-1913, 2015.
@article{LliNovTeiABC2015,
title = {Periodic solutions of Lienard differential equations via averaging theory of order two},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://dx.doi.org/10.1590/0001-3765201520140129},
doi = {10.1590/0001-3765201520140129},
year = {2015},
date = {2015-01-01},
journal = {Anais da Academia Brasileira de Ciências},
volume = {87},
number = {4},
pages = {1905-1913},
publisher = {FapUNIFESP (SciELO)},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
3.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
Corrigendum: Higher order averaging theory for finding periodic solutions via Brouwer degree (2014 Nonlinearity 27 563) Journal Article
Em: Nonlinearity, vol. 27, não 9, pp. 2417, 2014.
@article{LliNovTeiN2014c,
title = {Corrigendum: Higher order averaging theory for finding periodic solutions via Brouwer degree (2014 Nonlinearity 27 563)},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://dx.doi.org/10.1088/0951-7715/27/9/2417},
doi = {10.1088/0951-7715/27/9/2417},
year = {2014},
date = {2014-01-01},
journal = {Nonlinearity},
volume = {27},
number = {9},
pages = {2417},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
2.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
Higher order averaging theory for finding periodic solutions via Brouwer degree Journal Article
Em: Nonlinearity, vol. 27, não 3, pp. 563, 2014.
@article{LliNovTeiN2014,
title = {Higher order averaging theory for finding periodic solutions via Brouwer degree},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://dx.doi.org/10.1088/0951-7715/27/3/563},
doi = {10.1088/0951-7715/27/3/563},
year = {2014},
date = {2014-01-01},
journal = {Nonlinearity},
volume = {27},
number = {3},
pages = {563},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
1.
Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira
On the periodic solutions of a perturbed double pendulum Journal Article
Em: The São Paulo Journal of Mathematical Sciences, vol. 5, não 2, pp. 317, 2011.
@article{LliNovTeiSPJMS2011,
title = {On the periodic solutions of a perturbed double pendulum},
author = {Jaume Llibre and Douglas D. Novaes and Marco A. Teixeira},
url = {http://dx.doi.org/10.11606/issn.2316-9028.v5i2p317-330},
doi = {10.11606/issn.2316-9028.v5i2p317-330},
year = {2011},
date = {2011-01-01},
journal = {The São Paulo Journal of Mathematical Sciences},
volume = {5},
number = {2},
pages = {317},
publisher = {Universidade de Sao Paulo Sistema Integrado de Bibliotecas - SIBiUSP},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
