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**Artigos Selecionado**

**Artigos Selecionado**

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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, 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. |

Douglas D. Novaes On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set Journal Article Em: Physica D: Nonlinear Phenomena, 41 , pp. 133523, 2022. @article{Novaes2022, title = {On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set}, author = {Douglas D. Novaes}, url = {https://arxiv.org/abs/2201.02019}, doi = {10.1016/j.physd.2022.133523}, year = {2022}, date = {2022-09-06}, journal = {Physica D: Nonlinear Phenomena}, volume = {41}, pages = {133523}, abstract = {The second part of the Hilbert's sixteenth problem consists in determining the upper bound $CH(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $ngeq2$, it is still unknown whether $CH(n)$ is finite or not. The main achievements obtained so far establish lower bounds for $CH(n)$. Regarding asymptotic behavior, the best result says that $CH(n)$ grows as fast as $n^2log(n)$. Better lower bounds for small values of $n$ are known in the research literature. In the recent paper ``Some open problems in low dimensional dynamical systems'' by A. Gasull, Problem 18 proposes another Hilbert's sixteenth type problem, namely improving the lower bounds for $CL(n)$, $ninN$, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree $n$ can have. So far, $CL(n)geq [n/2],$ $ninN$, is the best known general lower bound. Again, better lower bounds for small values of $n$ are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that $CL(n)$ grows as fast as $n^2.$ This will be achieved by providing lower bounds for $CL(n)$, which improves every previous estimates for $ngeq 4$.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The second part of the Hilbert's sixteenth problem consists in determining the upper bound $CH(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $ngeq2$, it is still unknown whether $CH(n)$ is finite or not. The main achievements obtained so far establish lower bounds for $CH(n)$. Regarding asymptotic behavior, the best result says that $CH(n)$ grows as fast as $n^2log(n)$. Better lower bounds for small values of $n$ are known in the research literature. In the recent paper ``Some open problems in low dimensional dynamical systems'' by A. Gasull, Problem 18 proposes another Hilbert's sixteenth type problem, namely improving the lower bounds for $CL(n)$, $ninN$, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree $n$ can have. So far, $CL(n)geq [n/2],$ $ninN$, is the best known general lower bound. Again, better lower bounds for small values of $n$ are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that $CL(n)$ grows as fast as $n^2.$ This will be achieved by providing lower bounds for $CL(n)$, which improves every previous estimates for $ngeq 4$. |

Douglas D. Novaes; Leandro A. Silva On the non-existence of isochronous tangential centers in Filippov vector fields Journal Article Em: Proceedings of the American Mathematical Society, (1), pp. 10, 2022. @article{NovaesLSilva2022, title = {On the non-existence of isochronous tangential centers in Filippov vector fields}, author = {Douglas D. Novaes and Leandro A. Silva}, url = {https://arxiv.org/abs/2111.09020}, doi = {10.1090/proc/16047}, year = {2022}, date = {2022-07-01}, journal = {Proceedings of the American Mathematical Society}, number = {1}, pages = {10}, abstract = {The isochronicity problem is a classical problem in the qualitative theory of planar vector fields which consists in characterizing whether a center is isochronous or not, that is, if all the trajectories in a neighborhood of the center have the same period. This problem is usually investigated by means of the so-called period function. In this paper, we are interested in exploring the isochronicity problem for tangential centers of planar Filippov vector fields. By computing the period function for planar Filippov vector fields around tangential centers, we show that such centers are never isochronous. }, keywords = {}, pubstate = {published}, tppubtype = {article} } The isochronicity problem is a classical problem in the qualitative theory of planar vector fields which consists in characterizing whether a center is isochronous or not, that is, if all the trajectories in a neighborhood of the center have the same period. This problem is usually investigated by means of the so-called period function. In this paper, we are interested in exploring the isochronicity problem for tangential centers of planar Filippov vector fields. By computing the period function for planar Filippov vector fields around tangential centers, we show that such centers are never isochronous. |

Douglas D. Novaes An averaging result for periodic solutions of Carathéodory differential equations Journal Article Em: Proceedings of the American Mathematical Society, 150 (7), pp. 2945-2954, 2022. @article{Novaes2022, title = {An averaging result for periodic solutions of Carathéodory differential equations}, author = {Douglas D. Novaes}, url = {http://arxiv.org/abs/2108.01551}, doi = {10.1090/proc/15810}, year = {2022}, date = {2022-04-14}, journal = {Proceedings of the American Mathematical Society}, volume = {150}, number = {7}, pages = {2945-2954}, abstract = {This paper is concerned with the problem of existence of periodic solutions for perturbative Carathéodory differential equations. The main result provides sufficient conditions on the averaged equation that guarantee the existence of periodic solutions. Additional conditions are also provided to ensure the uniform convergence of a periodic solution to a constant function. The proof of the main theorem is mainly based on an abstract continuation result for operator equations.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This paper is concerned with the problem of existence of periodic solutions for perturbative Carathéodory differential equations. The main result provides sufficient conditions on the averaged equation that guarantee the existence of periodic solutions. Additional conditions are also provided to ensure the uniform convergence of a periodic solution to a constant function. The proof of the main theorem is mainly based on an abstract continuation result for operator equations. |

Douglas D. Novaes; Leandro A. Silva Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields Journal Article Em: Journal of Differential Equations, 300 (565), pp. 596, 2021. @article{NovaesLSilva2021, title = {Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields}, author = {Douglas D. Novaes and Leandro A. Silva}, url = {http://arxiv.org/abs/2010.00497}, doi = {10.1016/j.jde.2021.08.008}, year = {2021}, date = {2021-11-05}, journal = {Journal of Differential Equations}, volume = {300}, number = {565}, pages = {596}, abstract = {In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields Z=(Z+,Z-). More specifically, for positive integers k+ and k-, we consider a (2k+, 2k-)-monodromic tangential singularity, which is defined as an invisible contact of multiplicity 2k+ and 2k- between the discontinuity manifold and, respectively, the vectors fields Z+ and Z- for which a first-return map is well defined around the monodromic tangential singularity. We first prove that such a first-return map is analytic in a neighborhood of the monodromic tangential singularity. This allow us to define the Lyapunov coefficients. Then, as a consequence of a general property for pair of involutions, we obtain that the index of the first non-vanishing Lyapunov coefficient is always even. In addition, a recursive formula for computing all the Lyapunov coefficients is obtained. Such a formula is implemented in a Mathematica algorithm in the appendix. We also provide results regarding limit cycles bifurcating from monodromic tangential singularities. Several examples are analysed.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields Z=(Z+,Z-). More specifically, for positive integers k+ and k-, we consider a (2k+, 2k-)-monodromic tangential singularity, which is defined as an invisible contact of multiplicity 2k+ and 2k- between the discontinuity manifold and, respectively, the vectors fields Z+ and Z- for which a first-return map is well defined around the monodromic tangential singularity. We first prove that such a first-return map is analytic in a neighborhood of the monodromic tangential singularity. This allow us to define the Lyapunov coefficients. Then, as a consequence of a general property for pair of involutions, we obtain that the index of the first non-vanishing Lyapunov coefficient is always even. In addition, a recursive formula for computing all the Lyapunov coefficients is obtained. Such a formula is implemented in a Mathematica algorithm in the appendix. We also provide results regarding limit cycles bifurcating from monodromic tangential singularities. Several examples are analysed. |

Douglas D. Novaes; Gabriel A. Rondon Smoothing of nonsmooth systems near regular-tangential singularities and boundary limit cycle Journal Article Em: Nonlinearity, 34 (6), pp. 4202-4263, 2021. @article{NovRon2021, title = {Smoothing of nonsmooth systems near regular-tangential singularities and boundary limit cycle}, author = {Douglas D. Novaes and Gabriel A. Rondon}, url = {https://arxiv.org/abs/2003.09547}, doi = {10.1088/1361-6544/ac04be}, year = {2021}, date = {2021-05-25}, journal = {Nonlinearity}, volume = {34}, number = {6}, pages = {4202-4263}, abstract = {Understanding how tangential singularities evolve under smoothing processes was one of the first problem concerning regularization of Filippov systems. In this paper, we are interested in Cn-regularizations of Filippov systems around visible regular-tangential singularities of even multiplicity. More specifically, using Fenichel theory and blow-up methods, we aim to understand how the trajectories of the regularized system transits through the region of regularization. We apply our results to investigate Cn-regularizations of boundary limit cycles with even multiplicity contact with the switching manifold.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Understanding how tangential singularities evolve under smoothing processes was one of the first problem concerning regularization of Filippov systems. In this paper, we are interested in Cn-regularizations of Filippov systems around visible regular-tangential singularities of even multiplicity. More specifically, using Fenichel theory and blow-up methods, we aim to understand how the trajectories of the regularized system transits through the region of regularization. We apply our results to investigate Cn-regularizations of boundary limit cycles with even multiplicity contact with the switching manifold. |

Victoriano Carmona; Fernando Fernández Sánchez; Douglas D. Novaes A new simple proof for Lum-Chua's conjecture Journal Article Em: Nonlinear Analysis: Hybrid Systems, 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}, 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. |

Kamila S. Andrade; Oscar A. R. Cespedes; Dayane Cruz; Douglas D. Novaes Higher order Melnikov analysis of planar piecewise linear vector fields with nonlinear switching curve Journal Article Em: Journal of Differential Equations, 287 , pp. 1-36, 2021. @article{AndCesCruNov2021, title = {Higher order Melnikov analysis of planar piecewise linear vector fields with nonlinear switching curve}, author = {Kamila S. Andrade and Oscar A. R. Cespedes and Dayane Cruz and Douglas D. Novaes}, url = {https://arxiv.org/abs/2006.11352}, doi = {10.1016/j.jde.2021.03.039}, year = {2021}, date = {2021-03-29}, journal = {Journal of Differential Equations}, volume = {287}, pages = {1-36}, abstract = {In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)geq 4,$ $H(3)geq 8,$ $H(n)geq7,$ for $ngeq 4$ even, and $H(n)geq9,$ for $ngeq 5$ odd. This improves all the previous results for $ngeq2.$ Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov Theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)geq 4,$ $H(3)geq 8,$ $H(n)geq7,$ for $ngeq 4$ even, and $H(n)geq9,$ for $ngeq 5$ odd. This improves all the previous results for $ngeq2.$ Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov Theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold. |

Douglas D. Novaes; Francisco B.G. Silva Higher order analysis on the existence of periodic solutions in continuous differential equations via degree theory Journal Article Em: SIAM Journal on Mathematical Analysis, 53 (2), pp. 2476-2490, 2021. @article{NovGSilva2021, title = {Higher order analysis on the existence of periodic solutions in continuous differential equations via degree theory}, author = {Douglas D. Novaes and Francisco B.G. Silva}, url = {http://arxiv.org/abs/2006.10799}, year = {2021}, date = {2021-01-05}, journal = {SIAM Journal on Mathematical Analysis}, volume = {53}, number = {2}, pages = {2476-2490}, abstract = {Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of the Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the degree theory for operator equations, we perform a higher order analysis of continuous perturbed differential equations and derive sufficient conditions for the existence and uniform convergence of periodic solutions for such systems. We apply our results to study continuous non-Lipschitz higher order perturbations of a harmonic oscillator.}, keywords = {}, pubstate = {published}, tppubtype = {article} } Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of the Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the degree theory for operator equations, we perform a higher order analysis of continuous perturbed differential equations and derive sufficient conditions for the existence and uniform convergence of periodic solutions for such systems. We apply our results to study continuous non-Lipschitz higher order perturbations of a harmonic oscillator. |

Murilo R. Cândido; Douglas D. Novaes; Claudia Valls Periodic solutions and invariant torus in the Rössler System Journal Article Em: Nonlinearity, 33 (9), pp. 4512-4538, 2020. @article{CanNovVal2020, title = {Periodic solutions and invariant torus in the Rössler System}, author = {Murilo R. Cândido and Douglas D. Novaes and Claudia Valls}, url = {http://arxiv.org/abs/1903.02398}, doi = {10.1088/1361-6544/ab8bae}, year = {2020}, date = {2020-07-23}, journal = {Nonlinearity}, volume = {33}, number = {9}, pages = {4512-4538}, abstract = {The Rössler System is characterized by a three-parameter family of quadratic 3D vector fields. There exist two one-parameter families of Rössler Systems exhibiting a zero-Hopf equilibrium. For Rössler Systems near to one of these families, we provide generic conditions ensuring the existence of a torus bifurcation. In this case, the torus surrounds a periodic solution that bifurcates from the zero-Hopf equilibrium. For Rössler Systems near to the other family, we provide generic conditions for the existence of a periodic solution bifurcating from the zero-Hopf equilibrium. This improves currently known results regarding periodic solutions for such a family. In addition, the stability properties of the periodic solutions and invariant torus are analysed.}, keywords = {}, pubstate = {published}, tppubtype = {article} } The Rössler System is characterized by a three-parameter family of quadratic 3D vector fields. There exist two one-parameter families of Rössler Systems exhibiting a zero-Hopf equilibrium. For Rössler Systems near to one of these families, we provide generic conditions ensuring the existence of a torus bifurcation. In this case, the torus surrounds a periodic solution that bifurcates from the zero-Hopf equilibrium. For Rössler Systems near to the other family, we provide generic conditions for the existence of a periodic solution bifurcating from the zero-Hopf equilibrium. This improves currently known results regarding periodic solutions for such a family. In addition, the stability properties of the periodic solutions and invariant torus are analysed. |

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., 19 (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}, 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. |

Murilo R. Cândido; Douglas D. Novaes On the torus bifurcation in averaging theory Journal Article Em: Journal of Differential Equations, 268 (8), pp. 4555-4576, 2020. @article{CanNov2020, title = {On the torus bifurcation in averaging theory}, author = {Murilo R. Cândido and Douglas D. Novaes}, url = {https://arxiv.org/abs/1810.02992}, doi = {10.1016/j.jde.2019.10.031}, year = {2020}, date = {2020-01-31}, journal = {Journal of Differential Equations}, volume = {268}, number = {8}, pages = {4555-4576}, abstract = {In this paper, we take advantage of the averaging theory to investigate a torus bifurcation in two-parameter families of 2D nonautonomous differential equations. Our strategy consists in looking for generic conditions on the averaged functions that ensure the existence of a curve in the parameter space characterized by a Neimark-Sacker bifurcation in the corresponding Poincaré map. A Neimark-Sacker bifurcation for planar maps consists in the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. In addition, we apply our results to study a torus bifurcation in a family of 3D vector fields.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper, we take advantage of the averaging theory to investigate a torus bifurcation in two-parameter families of 2D nonautonomous differential equations. Our strategy consists in looking for generic conditions on the averaged functions that ensure the existence of a curve in the parameter space characterized by a Neimark-Sacker bifurcation in the corresponding Poincaré map. A Neimark-Sacker bifurcation for planar maps consists in the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. In addition, we apply our results to study a torus bifurcation in a family of 3D vector fields. |

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, 267 (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}, 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. |

Douglas D. Novaes; Marco A. Teixeira Shilnikov problem in Filippov dynamical systems Journal Article Em: Chaos, 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. |

Leonardo P. C. da Cruz; Douglas D. Novaes; Joan Torregrosa New lower bound for the Hilbert number in piecewise quadratic differential systems Journal Article Em: Journal of Differential Equations, 266 (7), pp. 4170-4203, 2019. @article{CruNovTor2018, title = {New lower bound for the Hilbert number in piecewise quadratic differential systems}, author = {Leonardo P. C. da Cruz and Douglas D. Novaes and Joan Torregrosa}, url = {http://dx.doi.org/10.1016/j.jde.2018.09.032 https://arxiv.org/abs/1809.03433}, doi = {10.1016/j.jde.2018.09.032}, year = {2019}, date = {2019-03-15}, journal = {Journal of Differential Equations}, volume = {266}, number = {7}, pages = {4170-4203}, abstract = {We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by $H_p(n)$ the extension of the Hilbert number to degree $n$ piecewise polynomial differential systems, then $H_p(2)geq 16.$ As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers.}, keywords = {}, pubstate = {published}, tppubtype = {article} } We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by $H_p(n)$ the extension of the Hilbert number to degree $n$ piecewise polynomial differential systems, then $H_p(2)geq 16.$ As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers. |

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, 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}, 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. |

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, 30 (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}, 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. |

Douglas D. Novaes; Joan Torregrosa On the extended Chebyshev systems with positive accuracy Journal Article Em: J. Math. Anal. Appl., 488 (1), pp. 171-186, 2017. @article{NovTor2017, title = {On the extended Chebyshev systems with positive accuracy}, author = {Douglas D. Novaes and Joan Torregrosa}, url = {http://dx.doi.org/10.1016/j.jmaa.2016.10.076}, doi = {10.1016/j.jmaa.2016.10.076}, year = {2017}, date = {2017-04-01}, journal = {J. Math. Anal. Appl.}, volume = {488}, number = {1}, pages = {171-186}, abstract = {A classical necessary condition for an ordered set of n+1 functions F to be an ECT-system in a closed interval is that all the Wronskians do not vanish. With this condition all the elements of Span(F) have at most n zeros taking into account the multiplicity. Here the problem of bounding the number of zeros of Span(F) is considered as well as the effectiveness of the upper bound when some Wronskians vanish. For this case we also study the possible configurations of zeros that can be realized by elements of Span(F). An application to counting the number of isolated periodic orbits for a family of nonsmooth systems is done. }, keywords = {}, pubstate = {published}, tppubtype = {article} } A classical necessary condition for an ordered set of n+1 functions F to be an ECT-system in a closed interval is that all the Wronskians do not vanish. With this condition all the elements of Span(F) have at most n zeros taking into account the multiplicity. Here the problem of bounding the number of zeros of Span(F) is considered as well as the effectiveness of the upper bound when some Wronskians vanish. For this case we also study the possible configurations of zeros that can be realized by elements of Span(F). An application to counting the number of isolated periodic orbits for a family of nonsmooth systems is done. |

Douglas D. Novaes; Gabriel Ponce; Régis Varão Chaos induced by sliding phenomena in Filippov systems Journal Article Em: Journal of Dynamics and Differential Equations, pp. 1-15, 2017. @article{NovPonVar2017, title = {Chaos induced by sliding phenomena in Filippov systems}, author = {Douglas D. Novaes and Gabriel Ponce and Régis Varão}, url = {http://dx.doi.org/10.1007/s10884-017-9580-8}, doi = {10.1007/s10884-017-9580-8}, year = {2017}, date = {2017-02-16}, journal = {Journal of Dynamics and Differential Equations}, pages = {1-15}, abstract = {In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit G. More specifically we prove that the first return map, defined nearby G, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each natural number m it has infinitely many periodic points with period m.}, keywords = {}, pubstate = {published}, tppubtype = {article} } In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit G. More specifically we prove that the first return map, defined nearby G, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each natural number m it has infinitely many periodic points with period m. |

Jaume Llibre; Ana C. Mereu; Douglas D. Novaes Averaging theory for discontinuous piecewise differential systems Journal Article Em: Journal of Differential Equations, 258 (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} } |

Douglas D. Novaes; Mike R. Jeffrey Regularization of hidden dynamics in piecewise smooth flows Journal Article Em: Journal of Differential Equations, 259 (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.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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. |

Jaume Llibre; Douglas D. Novaes; Marco A. Teixeira Higher order averaging theory for finding periodic solutions via Brouwer degree Journal Article Em: Nonlinearity, 27 (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} } |

Douglas D. Novaes On nonsmooth perturbations of nondegenerate planar centers Journal Article Em: Publicacions Matemàtiques, 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}, journal = {Publicacions Matemàtiques}, volume = {EXTRA}, pages = {395-420}, publisher = {Universitat Autonoma de Barcelona}, keywords = {}, pubstate = {published}, tppubtype = {article} } |

**Formação**

**Formação**

Livre-Docente em Matemática, 2019, UNICAMP.

Pós-Doutorado, 2015 e 2016, UNICAMP e UAB.

Doutor em Matemática, 2015, UNICAMP (download da tese).

Estágio de Pesquisa no Exterior, 2014, UAB, Barcelona-Espanha.

Mestre em Matemática, 2012, UNICAMP (link da dissertação).

Licenciatura em Matemática, 2010, UNICAMP.

**Prêmios**

**Prêmios**

Menção Honrosa – Prêmio Carlos Teobaldo Gutierrez Vidalon 2016.

Menção Honrosa – XVIII Congresso Interno de Iniciação Científica da UNICAMP.

**Interesses Científicos**

**Interesses Científicos**

##### (clique aqui para mais detalhes)

- Averaging theory, Melnikov method, Lyapunov-Schmidit reduction, Relaxation Oscillation theory, and other tools to study and detect invariant sets.
- Chebyshev systems with positive accuracy and their applications in dynamics.
- Hidden dynamics, regularization and pinching of Filippov and non Filippov systems.
- Singular perturbation problems and their relation with regularization of piecewise smooth vector fields.
- Typical cycles and global dynamics of piecewise smooth vector fields.
- Sliding dynamics and chaos.
- Invariant measures for piecewise smooth vector fields.
- Differential Inclusions.

No arquivo a seguir você encontrará alguns comentários elaborados pelo Prof. Marco Antonio Teixeira sobre uma das linhas de pesquisa na qual estou inserido: Non-smooth dynamical system – Reflections and Guidelines.

**Próximos Eventos**

**Próximos Eventos**

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