2016
16.
Márcio R. A. Gouveia; Jaume Llibre; Douglas D. Novaes; Cláudio Pessoa
Piecewise smooth dynamical systems: persistenc of periodic solutions and normal forms Journal Article
Em: Journal of Differential Equations, vol. 260, pp. 6180-6129, 2016.
@article{GouLliNovPesJDE2015,
title = {Piecewise smooth dynamical systems: persistenc of periodic solutions and normal forms},
author = {Márcio R. A. Gouveia and Jaume Llibre and Douglas D. Novaes and Cláudio Pessoa},
url = {http://dx.doi.org/10.1016/j.jde.2015.12.034},
doi = {10.1016/j.jde.2015.12.034},
year = {2016},
date = {2016-04-05},
journal = {Journal of Differential Equations},
volume = {260},
pages = {6180-6129},
abstract = {We consider a n-dimensional piecewise smooth vector fields with two zones separated by a hyperplane S which admits an invariant hyperplane O transversal to S containing a period annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n=3 we provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.
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We consider a n-dimensional piecewise smooth vector fields with two zones separated by a hyperplane S which admits an invariant hyperplane O transversal to S containing a period annulus A fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in A. When n=3 we provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.
2015
15.
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.
},
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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.
14.
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.},
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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.
13.
Jaume Llibre; Ana C. Mereu; Douglas D. Novaes
Averaging theory for discontinuous piecewise differential systems Journal Article
Em: Journal of Differential Equations, vol. 258, não 11, pp. 4007 - 4032, 2015.
@article{LliMerNovJDF2015,
title = {Averaging theory for discontinuous piecewise differential systems},
author = {Jaume Llibre and Ana C. Mereu and Douglas D. Novaes},
url = {http://dx.doi.org/10.1016/j.jde.2015.01.022},
doi = {10.1016/j.jde.2015.01.022},
year = {2015},
date = {2015-01-01},
journal = {Journal of Differential Equations},
volume = {258},
number = {11},
pages = {4007 - 4032},
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12.
Douglas D. Novaes; Enirque Ponce
A Simple Solution to the Braga–Mello Conjecture Journal Article
Em: International Journal of Bifurcation and Chaos, vol. 25, não 01, pp. 1550009, 2015.
@article{NovPonJBC2015,
title = {A Simple Solution to the Braga–Mello Conjecture},
author = {Douglas D. Novaes and Enirque Ponce},
url = {http://dx.doi.org/10.1142/S0218127415500091},
doi = {10.1142/S0218127415500091},
year = {2015},
date = {2015-01-01},
journal = {International Journal of Bifurcation and Chaos},
volume = {25},
number = {01},
pages = {1550009},
abstract = {Recently Braga and Mello conjectured that for a given natural number n there is a piecewise linear system with two zones in the plane with exactly n limit cycles. In this paper, we prove a result from which the conjecture is an immediate consequence. Several explicit examples are given where location and stability of limit cycles are provided.},
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Recently Braga and Mello conjectured that for a given natural number n there is a piecewise linear system with two zones in the plane with exactly n limit cycles. In this paper, we prove a result from which the conjecture is an immediate consequence. Several explicit examples are given where location and stability of limit cycles are provided.
11.
Jaume Llibre; Douglas D. Novaes
Improving the averaging theory for computing periodic solutions of the differential equations Journal Article
Em: Zeitschrift für angewandte Mathematik und Physik, vol. 66, não 4, pp. 1401-1412, 2015.
@article{LliNovZAMP2015,
title = {Improving the averaging theory for computing periodic solutions of the differential equations},
author = {Jaume Llibre and Douglas D. Novaes},
url = {http://dx.doi.org/10.1007/s00033-014-0460-3},
doi = {10.1007/s00033-014-0460-3},
year = {2015},
date = {2015-01-01},
journal = {Zeitschrift für angewandte Mathematik und Physik},
volume = {66},
number = {4},
pages = {1401-1412},
publisher = {Springer Basel},
keywords = {},
pubstate = {published},
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10.
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.},
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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.
9.
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}
}
8.
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.},
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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.
7.
Márcio R. A. Gouveia; Jaume Llibre; Douglas D. Novaes
On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems Journal Article
Em: Applied Mathematics and Computation, vol. 271, pp. 365 - 374, 2015.
@article{GouLliNovAPC2015,
title = {On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems},
author = {Márcio R. A. Gouveia and Jaume Llibre and Douglas D. Novaes},
url = {http://dx.doi.org/10.1016/j.amc.2015.09.022},
doi = {10.1016/j.amc.2015.09.022},
year = {2015},
date = {2015-01-01},
journal = {Applied Mathematics and Computation},
volume = {271},
pages = {365 - 374},
abstract = {Abstract In this paper we consider the linear differential center (x',y') = (− y , x ) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0 . Using the Bendixson transformation we provide sufficient conditions to ensure the existence of a crossing limit cycle coming purely from the infinity. We also study the displacement function for a class of discontinuous piecewise smooth differential system.},
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Abstract In this paper we consider the linear differential center (x',y') = (− y , x ) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0 . Using the Bendixson transformation we provide sufficient conditions to ensure the existence of a crossing limit cycle coming purely from the infinity. We also study the displacement function for a class of discontinuous piecewise smooth differential system.
6.
Douglas D. Novaes; Mike R. Jeffrey
Regularization of hidden dynamics in piecewise smooth flows Journal Article
Em: Journal of Differential Equations, vol. 259, não 9, pp. 4615 - 4633, 2015.
@article{NovJefJDE2015,
title = {Regularization of hidden dynamics in piecewise smooth flows},
author = {Douglas D. Novaes and Mike R. Jeffrey},
url = {http://dx.doi.org/10.1016/j.jde.2015.06.005},
doi = {10.1016/j.jde.2015.06.005},
year = {2015},
date = {2015-01-01},
journal = {Journal of Differential Equations},
volume = {259},
number = {9},
pages = {4615 - 4633},
abstract = {This paper studies the equivalence between differentiable and non-differentiable dynamics in Rn . Filippov's theory of discontinuous differential equations allows us to find flow solutions of dynamical systems whose vector fields undergo switches at thresholds in phase space. The canonical convex combination at the discontinuity is only the linear part of a nonlinear combination that more fully explores Filippov's most general problem: the differential inclusion. Here we show how recent work relating discontinuous systems to singular limits of continuous (or regularized) systems extends to nonlinear combinations. We show that if sliding occurs in a discontinuous systems, there exists a differentiable slow–fast system with equivalent slow invariant dynamics. We also show the corresponding result for the pinching method, a converse to regularization which approximates a smooth system by a discontinuous one.},
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This paper studies the equivalence between differentiable and non-differentiable dynamics in Rn . Filippov's theory of discontinuous differential equations allows us to find flow solutions of dynamical systems whose vector fields undergo switches at thresholds in phase space. The canonical convex combination at the discontinuity is only the linear part of a nonlinear combination that more fully explores Filippov's most general problem: the differential inclusion. Here we show how recent work relating discontinuous systems to singular limits of continuous (or regularized) systems extends to nonlinear combinations. We show that if sliding occurs in a discontinuous systems, there exists a differentiable slow–fast system with equivalent slow invariant dynamics. We also show the corresponding result for the pinching method, a converse to regularization which approximates a smooth system by a discontinuous one.
2014
5.
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},
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4.
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},
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3.
Douglas D. Novaes
On nonsmooth perturbations of nondegenerate planar centers Journal Article
Em: Publicacions Matemàtiques, vol. EXTRA, pp. 395-420, 2014.
@article{NovPM2014,
title = {On nonsmooth perturbations of nondegenerate planar centers},
author = {Douglas D. Novaes},
url = {http://dx.doi.org/10.5565/publmat_extra14_20},
doi = {10.5565/publmat_extra14_20},
year = {2014},
date = {2014-01-01},
urldate = {2014-01-01},
journal = {Publicacions Matemàtiques},
volume = {EXTRA},
pages = {395-420},
publisher = {Universitat Autonoma de Barcelona},
keywords = {},
pubstate = {published},
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2013
2.
Douglas D. Novaes
Perturbed damped pendulum: finding periodic solutions via averaging method Journal Article
Em: Revista Brasileira de Ensino de Física, vol. 35, não 1, pp. 01-07, 2013.
@article{NovaesRBEF2013,
title = {Perturbed damped pendulum: finding periodic solutions via averaging method},
author = {Douglas D. Novaes},
url = {http://dx.doi.org/10.1590/s1806-11172013000100014},
doi = {10.1590/s1806-11172013000100014},
year = {2013},
date = {2013-01-01},
journal = {Revista Brasileira de Ensino de Física},
volume = {35},
number = {1},
pages = {01-07},
publisher = {FapUNIFESP (SciELO)},
keywords = {},
pubstate = {published},
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2011
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},
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