**Preprints**

Douglas D. Novaes; Luan V. M. F. Silva

A Melnikov analysis on a class of second order discontinuous differential equations Journal Article

Em: Preprint, 2023.

@article{NovLSilva2023b,

title = {A Melnikov analysis on a class of second order discontinuous differential equations},

author = {Douglas D. Novaes and Luan V. M. F. Silva},

url = {https://arxiv.org/abs/2312.02738},

year = {2023},

date = {2023-12-09},

urldate = {2023-12-09},

journal = {Preprint},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Leonardo F. Cavenaghi; Lino Grama; Ricardo M. Martins; Douglas D. Novaes

The complete dynamics description of positively curved metrics in the Wallach flag manifold SU(3)/T2 Journal Article

Em: Preprint, 2023.

@article{CavGraMarNov23,

title = {The complete dynamics description of positively curved metrics in the Wallach flag manifold SU(3)/T2},

author = {Leonardo F. Cavenaghi and Lino Grama and Ricardo M. Martins and Douglas D. Novaes },

url = {http://arxiv.org/abs/2307.06418},

year = {2023},

date = {2023-07-13},

journal = {Preprint},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Matheus G. C. Cunha; Douglas D. Novaes; Gabriel Ponce

On the Hausdorff dimension of sliding Shilnikov invariant sets Journal Article

Em: Preprint, 2023.

@article{CunNovPon2023,

title = {On the Hausdorff dimension of sliding Shilnikov invariant sets},

author = {Matheus G. C. Cunha and Douglas D. Novaes and Gabriel Ponce},

url = {https://arxiv.org/abs/2312.10720},

year = {2023},

date = {2023-05-20},

urldate = {2023-05-20},

journal = {Preprint},

abstract = {In this paper, we conduct a local analysis of the first return map defined on the invariant set $Lambda$ induced by a sliding Shilnikov connection, using the theory of conformal iterated function systems. As a result, we calculate its Hausdorff dimension and Lebesgue measure, obtaining the results $0keywords = {},

pubstate = {published},

tppubtype = {article}

}

Douglas D. Novaes; Leandro A. Silva

On the cyclicity of monodromic tangential singularities: a look beyond the pseudo-Hopf bifurcation Journal Article

Em: Preprint, 2023.

@article{NovaesLSilva2023,

title = {On the cyclicity of monodromic tangential singularities: a look beyond the pseudo-Hopf bifurcation},

author = {Douglas D. Novaes and Leandro A. Silva},

url = {https://arxiv.org/abs/2303.06027},

year = {2023},

date = {2023-03-10},

journal = {Preprint},

abstract = {The cyclicity problem consists in estimating the number of limit cycles bifurcating from a monodromic singularity of planar vector fields and is usually addressed by means of Lyapunov coefficients. For nonsmooth systems, besides the limit cycles bifurcating by varying the Lyapunov coefficients, monodromic singularities lying on the switching curve can always be split apart generating, under suitable conditions, a sliding region and an extra limit cycle surrounding it. This bifurcation phenomenon is called pseudo-Hopf bifurcation and has been used to increase the lower bounds for the cyclicity of monodromic singularities in Filippov vector fields. In this paper, we aim to go beyond the pseudo-Hopf bifurcation by showing that the destruction of (2k,2k)-monodromic tangential singularities give birth to at least k limit cycles surrounding sliding segments.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Joyce A. Casimiro; Ricardo M. Martins; Douglas D. Novaes

Poincaré-Hopf Theorem for Filippov vector fields on 2 dimensional manifolds Journal Article

Em: Preprint, 2023.

@article{CasMarNov23,

title = {Poincaré-Hopf Theorem for Filippov vector fields on 2 dimensional manifolds},

author = {Joyce A. Casimiro and Ricardo M. Martins and Douglas D. Novaes},

url = {https://arxiv.org/abs/2303.04316},

year = {2023},

date = {2023-03-08},

journal = {Preprint},

abstract = {The Euler characteristic of a 2-dimensional compact manifold and the local behavior of smooth vector fields defined on it are related to each other by means of the Poincaré-Hopf Theorem. Despite of the importance of Filippov vector fields, concerning both their theoretical and applied aspects, until now, it was not known if this result it still true for Filippov vector fields. In this paper, we show it is.

While in the smooth case the singularities consist of the points where the vector field vanishes, in the context of Filippov vector fields the notion of singularity also comprehend new kinds of points, namely, pseudo-equilibria and tangency points. Here, the classical index definition for singularities of smooth vector fields is extended to singularities of Filippov vector fields. Such an extension is based on an invariance property under a regularization process. With this new index definition, we provide a version of the Poincaré-Hopf Theorem for Filippov vector fields. Consequently, we also get a Hairy Ball Theorem in this context, i.e. "any Filippov vector field defined on a sphere must have at least one singularity (in the Filippov sense)".},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

While in the smooth case the singularities consist of the points where the vector field vanishes, in the context of Filippov vector fields the notion of singularity also comprehend new kinds of points, namely, pseudo-equilibria and tangency points. Here, the classical index definition for singularities of smooth vector fields is extended to singularities of Filippov vector fields. Such an extension is based on an invariance property under a regularization process. With this new index definition, we provide a version of the Poincaré-Hopf Theorem for Filippov vector fields. Consequently, we also get a Hairy Ball Theorem in this context, i.e. "any Filippov vector field defined on a sphere must have at least one singularity (in the Filippov sense)".

Murilo R. Cândido; Douglas D. Novaes

On the stability of smooth branches of periodic solutions for higher order perturbed differential systems Journal Article

Em: Preprint, 2022.

@article{CanNov2022,

title = {On the stability of smooth branches of periodic solutions for higher order perturbed differential systems},

author = {Murilo R. Cândido and Douglas D. Novaes},

url = {http://arxiv.org/abs/2212.11812},

year = {2022},

date = {2022-12-22},

journal = {Preprint},

abstract = {The averaging method combined with the Lyapunov-Schmidt reduction provides sufficient conditions for the existence of periodic solutions of the following class of perturbative $T$-periodic nonautonomous differential equations $x'=F_0(t,x)+varepsilon F(t,x,varepsilon)$. Such periodic solutions bifurcate from a manifold $mathcal{Z}$ of periodic solutions of the unperturbed system $x'=F_0(t,x)$. Determining the stability of this kind of periodic solutions involves the computation of eigenvalues of matrix-valued functions $M(varepsilon)$, which can done using the theory of $k$-hyperbolic matrices. Usually, in this theory, a diagonalizing process of $k$-jets of $M(varepsilon)$ must be employed and no general algorithm exists for doing that. In this paper, we develop an alternative strategy for determining the stability of the periodic solutions without the need of such a diagonalization process, which can work even when the diagonalization is not possible. Applications of our result for two families of $4$D vector fields are also presented.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Douglas D. Novaes; Pedro C. Pereira

On the number of isolated invariant tori for 3D polynomial vector fields Journal Article

Em: Preprint, 2022.

@article{NovPer2022b,

title = {On the number of isolated invariant tori for 3D polynomial vector fields},

author = {Douglas D. Novaes and Pedro C. Pereira},

url = {https://arxiv.org/abs/2212.12006},

year = {2022},

date = {2022-12-22},

journal = {Preprint},

abstract = {The 16th Hilbert's Problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree $m$, has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, $3$D polynomial vector fields of a given degree $m$. Here, as an extension of such a problem in the $3$D space, we investigate the number of isolated invariant tori in $3$D polynomial vector fields. In this context, given a natural number $m$, we denote by $N(m)$ the upper bound for the number of isolated invariant tori of $3$D polynomial vector fields of degree $m$. Our main result provides a lower bound for $N(m)$ in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree $[m/2]-1$.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Tiago de Carvalho; Douglas D. Novaes; Durval J. Tonon

Sliding motion on tangential sets of Filippov systems Journal Article

Em: Preprint, 2021.

@article{CarNovTon2021,

title = {Sliding motion on tangential sets of Filippov systems},

author = {Tiago de Carvalho and Douglas D. Novaes and Durval J. Tonon},

url = {https://arxiv.org/abs/2111.12377},

year = {2021},

date = {2021-11-25},

journal = {Preprint},

abstract = {We consider piecewise smooth vector fields $Z=(Z_+, Z_-)$ defined in $R^n$ where both vector fields are tangent to the switching manifold $Sigma$ along a manifold $M$. Our main purpose is to study the existence of an invariant vector field defined on $M$, that we call {it tangential sliding vector field}. We provide necessary and sufficient conditions under $Z$ to characterize the existence of this vector field. Considering a regularization process, we proved that this tangential sliding vector field is topological equivalent to an invariant dynamics under the slow manifold. Besides then, at the end of the work we apply the results in a system that governs the intermittent treatment of HIV.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

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}

}

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.

**Forthcoming**

Douglas D. Novaes

Regularization of sliding Shilnikov connections: robustness of chaos under smoothing process Journal Article

Em: Forthcoming, 2018.

@article{Novaes2018,

title = {Regularization of sliding Shilnikov connections: robustness of chaos under smoothing process},

author = {Douglas D. Novaes},

year = {2018},

date = {2018-08-30},

journal = {Forthcoming},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Douglas D. Novaes; Enirque Ponce

Existence of periodic solution in nonsmooth Lienard differential systems via comparison method Journal Article

Em: Forthcoming, 2016.

@article{NovPon2016,

title = {Existence of periodic solution in nonsmooth Lienard differential systems via comparison method},

author = {Douglas D. Novaes and Enirque Ponce},

year = {2016},

date = {2016-07-16},

journal = {Forthcoming},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

Paulo R. Guimarães; Douglas D. Novaes; Sérgio F. Reis; Marco A. Teixeira

The complexity-stability debate and May's paradox: A perspective from bifurcation theory Journal Article

Em: Forthcoming, 2016.

@article{GuiNovReiTei2016,

title = {The complexity-stability debate and May's paradox: A perspective from bifurcation theory},

author = {Paulo R. Guimarães and Douglas D. Novaes and Sérgio F. Reis and Marco A. Teixeira},

year = {2016},

date = {2016-01-01},

journal = {Forthcoming},

keywords = {},

pubstate = {published},

tppubtype = {article}

}