21.
Jaume Llibre; Douglas D. Novaes; Claudia Valls
Existence of a cylinder foliated by periodic orbits in the generalized Chazy differential equation Journal Article
Em: Chaos, vol. 33, iss. 7, pp. 073104, 2023.
@article{LliNovVal23,
title = {Existence of a cylinder foliated by periodic orbits in the generalized Chazy differential equation},
author = {Jaume Llibre and Douglas D. Novaes and Claudia Valls},
url = {https://arxiv.org/abs/2307.00087},
doi = {10.1063/5.0138309},
year = {2023},
date = {2023-07-05},
urldate = {2023-07-05},
journal = {Chaos},
volume = {33},
issue = {7},
pages = {073104},
abstract = {The generalized Chazy differential equation corresponds to the following $2$-parameter family of differential equations
$$
dddot x+|x|^q ddot x+dfrac{k |x|^q}{x}dot x^2=0,
$$
which has its regularity varying with $q$ , a positive integer. Indeed, for $q=1$ it is discontinuous on the straight line $x=0$, whereas for $q$ a positive even integer it is polynomial, and for $q>1$ a positive odd integer it is continuous but not differentiable on the straight line $x=0$. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for $q=2$ and $k=3$. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for $k=q+1$ and any positive integer $q$ , has actually an invariant topological cylinder foliated by periodic solutions in the $(x,dot x,ddot x)$-space . In order to set forth the bases of our approach, we start by considering $q=1,2,3$, which are representatives of the different classes of regularity. For an arbitrary positive integer $q$ , an algorithm is provided for checking the sufficient conditions for the existence of such invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to $q=100$.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The generalized Chazy differential equation corresponds to the following $2$-parameter family of differential equations
$$
dddot x+|x|^q ddot x+dfrac{k |x|^q}{x}dot x^2=0,
$$
which has its regularity varying with $q$ , a positive integer. Indeed, for $q=1$ it is discontinuous on the straight line $x=0$, whereas for $q$ a positive even integer it is polynomial, and for $q>1$ a positive odd integer it is continuous but not differentiable on the straight line $x=0$. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for $q=2$ and $k=3$. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for $k=q+1$ and any positive integer $q$ , has actually an invariant topological cylinder foliated by periodic solutions in the $(x,dot x,ddot x)$-space . In order to set forth the bases of our approach, we start by considering $q=1,2,3$, which are representatives of the different classes of regularity. For an arbitrary positive integer $q$ , an algorithm is provided for checking the sufficient conditions for the existence of such invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to $q=100$.
$$
dddot x+|x|^q ddot x+dfrac{k |x|^q}{x}dot x^2=0,
$$
which has its regularity varying with $q$ , a positive integer. Indeed, for $q=1$ it is discontinuous on the straight line $x=0$, whereas for $q$ a positive even integer it is polynomial, and for $q>1$ a positive odd integer it is continuous but not differentiable on the straight line $x=0$. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for $q=2$ and $k=3$. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for $k=q+1$ and any positive integer $q$ , has actually an invariant topological cylinder foliated by periodic solutions in the $(x,dot x,ddot x)$-space . In order to set forth the bases of our approach, we start by considering $q=1,2,3$, which are representatives of the different classes of regularity. For an arbitrary positive integer $q$ , an algorithm is provided for checking the sufficient conditions for the existence of such invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to $q=100$.
20.
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.
19.
Jaume Llibre; Douglas D. Novaes; Camila A. B. Rodrigues
Bifurcations from families of periodic solutions in piecewise differential systems Journal Article
Em: Physica D, vol. 404, pp. 132342, 2020.
@article{LliNovRod2020,
title = {Bifurcations from families of periodic solutions in piecewise differential systems},
author = {Jaume Llibre and Douglas D. Novaes and Camila A. B. Rodrigues},
url = {https://arxiv.org/abs/1804.08175},
doi = {10.1016/j.physd.2020.132342},
year = {2020},
date = {2020-01-28},
journal = {Physica D},
volume = {404},
pages = {132342},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
18.
Jaume Llibre; Douglas D. Novaes; Iris O. Zeli
Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems Journal Article
Em: Revista de Matemátca Iberoamericana, vol. 36, pp. 291-318, 2020.
@article{LliNovZel2019,
title = {Limit cycles of piecewise polynomial perturbations of higher dimensional linear differential systems},
author = {Jaume Llibre and Douglas D. Novaes and Iris O. Zeli},
url = {hyyp://dx.doi.org/10.4171/rmi/1131
https://arxiv.org/abs/1801.01730},
doi = {10.4171/rmi/1131},
year = {2020},
date = {2020-01-01},
urldate = {2020-01-01},
journal = {Revista de Matemátca Iberoamericana},
volume = {36},
pages = {291-318},
abstract = {The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non--autonomous $n$-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold $CZsubsetR^n$ of periodic solutions satisfying $dim(mathcal{Z})
a class of discontinuous piecewise polynomial differential systems
with two zones. More precisely, we study the periodic solutions of
the following differential system
[
x'=Mx+ e F_1^n(x)+e^2F_2^n(x),
]
in $R^{d+2}$ where $e$ is a small parameter, $M$ is a
$(d+2)times(d+2)$ matrix having one pair of pure imaginary
conjugate eigenvalues, $m$ zeros eigenvalues, and $d-m$ non--zero real eigenvalues.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The averaging theory has been extensively employed for studying periodic solutions of smooth and nonsmooth differential systems. Here, we extend the averaging theory for studying periodic solutions a class of regularly perturbed non--autonomous $n$-dimensional discontinuous piecewise smooth differential system. As a fundamental hypothesis, it is assumed that the unperturbed system has a manifold $CZsubsetR^n$ of periodic solutions satisfying $dim(mathcal{Z})<n.$ Then, we apply this result to study limit cycles bifurcating from periodic solutions of
linear differential systems, $x'=Mx$, when they are perturbed inside
a class of discontinuous piecewise polynomial differential systems
with two zones. More precisely, we study the periodic solutions of
the following differential system
[
x'=Mx+ e F_1^n(x)+e^2F_2^n(x),
]
in $R^{d+2}$ where $e$ is a small parameter, $M$ is a
$(d+2)times(d+2)$ matrix having one pair of pure imaginary
conjugate eigenvalues, $m$ zeros eigenvalues, and $d-m$ non--zero real eigenvalues.
linear differential systems, $x'=Mx$, when they are perturbed inside
a class of discontinuous piecewise polynomial differential systems
with two zones. More precisely, we study the periodic solutions of
the following differential system
[
x'=Mx+ e F_1^n(x)+e^2F_2^n(x),
]
in $R^{d+2}$ where $e$ is a small parameter, $M$ is a
$(d+2)times(d+2)$ matrix having one pair of pure imaginary
conjugate eigenvalues, $m$ zeros eigenvalues, and $d-m$ non--zero real eigenvalues.
17.
Jéfferson L. R. Bastos; Claudio A. Buzzi; Jaume Llibre; Douglas D. Novaes
Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold Journal Article
Em: Journal of Differential Equations, vol. 267, não 5, pp. 3748-3767, 2019.
@article{BBLN2019,
title = {Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold},
author = {Jéfferson L. R. Bastos and Claudio A. Buzzi and Jaume Llibre and Douglas D. Novaes},
url = {http://dx.doi.org/10.1016/j.jde.2019.04.019
https://arxiv.org/abs/1810.02993},
doi = {10.1016/j.jde.2019.04.019},
year = {2019},
date = {2019-09-05},
urldate = {2019-09-05},
journal = {Journal of Differential Equations},
volume = {267},
number = {5},
pages = {3748-3767},
abstract = {We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.
16.
Jackson Itikawa; Jaume Llibre; Douglas D. Novaes
A new result on averaging theory for a class of discontinuous planar differential systems with applications Journal Article
Em: Rev. Mat. Iberoam., vol. 33, não 4, pp. 1247-1265, 2017.
@article{ItiLliNov2017,
title = {A new result on averaging theory for a class of discontinuous planar differential systems with applications},
author = {Jackson Itikawa and Jaume Llibre and Douglas D. Novaes},
url = {dx.doi.or/10.4171/rmi/970},
doi = {10.4171/rmi/970},
year = {2017},
date = {2017-11-17},
journal = {Rev. Mat. Iberoam.},
volume = {33},
number = {4},
pages = {1247-1265},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
15.
Murilo R. Cândido; Jaume Llibre; Douglas D. Novaes
Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction Journal Article
Em: Nonlinearity, vol. 30, não 9, pp. 3560-3586, 2017.
@article{CanLliNov2016,
title = {Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction},
author = {Murilo R. Cândido and Jaume Llibre and Douglas D. Novaes},
url = {http://doi.org/10.1088/1361-6544/aa7e95
https://arxiv.org/abs/1611.04807},
doi = {10.1088/1361-6544/aa7e95},
year = {2017},
date = {2017-08-14},
urldate = {2017-08-14},
journal = {Nonlinearity},
volume = {30},
number = {9},
pages = {3560-3586},
abstract = {In this work we first provide sufficient conditions to assure the persistence of some zeros of perturbative functions. Then we use this result to compute the bifurcation functions which allow us to study the periodic solutions of T-periodic smooth differential system. It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this work we first provide sufficient conditions to assure the persistence of some zeros of perturbative functions. Then we use this result to compute the bifurcation functions which allow us to study the periodic solutions of T-periodic smooth differential system. It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.
14.
Jaume Llibre; Douglas D. Novaes; Camila A. B. Rodrigues
Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones Journal Article
Em: Physica D: Nonlinear Phenomena, 2017.
@article{LliNovRod2017,
title = {Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones},
author = {Jaume Llibre and Douglas D. Novaes and Camila A. B. Rodrigues},
url = {http://dx.doi.org/10.1016/j.physd.2017.05.003},
doi = {10.1016/j.physd.2017.05.003},
year = {2017},
date = {2017-05-25},
journal = {Physica D: Nonlinear Phenomena},
abstract = {This work is devoted to study the existence of periodic solutions for a class of discontinuous differential systems with many zones. We show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class. We also provide some examples dealing with nonsmooth perturbations of nonlinear centers.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
This work is devoted to study the existence of periodic solutions for a class of discontinuous differential systems with many zones. We show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class. We also provide some examples dealing with nonsmooth perturbations of nonlinear centers.
13.
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.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
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.
12.
Jaume Llibre; Douglas D. Novaes
On the continuation of periodic solutions in discontinuous dynamical systems Journal Article
Em: 2015.
@article{LliNov2016,
title = {On the continuation of periodic solutions in discontinuous dynamical systems},
author = {Jaume Llibre and Douglas D. Novaes},
url = {http://arxiv.org/abs/1504.03008},
year = {2015},
date = {2015-12-31},
abstract = {Motivated by problems coming from different areas of the applied science we study the periodic solutions of the following differential system x'(t)=F0(t,x)+e F1(t,x)+e^2 R(t,x,e),
when F0, F1, and R are discontinuous piecewise functions, and e is a small parameter. It is assumed that the manifold Z of all periodic solutions of the unperturbed system x'=F0(t,x) has dimension n or smaller then n. The averaging theory is one of the best tools to attack this problem. This theory is completely developed when F0, F1 and R are continuous functions, and
also when F0=0 for a class of discontinuous differential systems. Nevertheless does not exist the averaging theory for studying the periodic solutions of discontinuous differential system when
F0 is different from 0. In this paper we develop this theory for a big class of discontinuous differential systems.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Motivated by problems coming from different areas of the applied science we study the periodic solutions of the following differential system x'(t)=F0(t,x)+e F1(t,x)+e^2 R(t,x,e),
when F0, F1, and R are discontinuous piecewise functions, and e is a small parameter. It is assumed that the manifold Z of all periodic solutions of the unperturbed system x'=F0(t,x) has dimension n or smaller then n. The averaging theory is one of the best tools to attack this problem. This theory is completely developed when F0, F1 and R are continuous functions, and
also when F0=0 for a class of discontinuous differential systems. Nevertheless does not exist the averaging theory for studying the periodic solutions of discontinuous differential system when
F0 is different from 0. In this paper we develop this theory for a big class of discontinuous differential systems.
when F0, F1, and R are discontinuous piecewise functions, and e is a small parameter. It is assumed that the manifold Z of all periodic solutions of the unperturbed system x'=F0(t,x) has dimension n or smaller then n. The averaging theory is one of the best tools to attack this problem. This theory is completely developed when F0, F1 and R are continuous functions, and
also when F0=0 for a class of discontinuous differential systems. Nevertheless does not exist the averaging theory for studying the periodic solutions of discontinuous differential system when
F0 is different from 0. In this paper we develop this theory for a big class of discontinuous differential systems.
11.
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.
10.
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.
9.
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},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
8.
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},
tppubtype = {article}
}
7.
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.
6.
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.
5.
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)},
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4.
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.
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},
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tppubtype = {article}
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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},
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pubstate = {published},
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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}
}
