Preprints
Murilo R. Cândido; Douglas D. Novaes; Nasrin Sadri
Invariant Tori and Periodic Orbits in the FitzHugh-Nagumo System Journal Article
Em: Preprint, 2024.
@article{CanNovSad2024,
title = {Invariant Tori and Periodic Orbits in the FitzHugh-Nagumo System},
author = {Murilo R. Cândido and Douglas D. Novaes and Nasrin Sadri},
url = {https://arxiv.org/abs/2408.12771},
year = {2024},
date = {2024-07-14},
journal = {Preprint},
abstract = {The FitzHugh-Nagumo system is a $4$-parameter family of $3$D vector field used for modeling neural excitation and nerve impulse propagation. The origin represents a Hopf-zero equilibrium in the FitzHugh-Nagumo system for two classes of parameters. In this paper, we employ recent techniques in averaging theory to investigate, besides periodic solutions, the bifurcation of invariant tori within the aforementioned families. We provide explicit generic conditions for the existence of these tori and analyze their stability properties.
Furthermore, we employ the backward differentiation formula to solve the stiff differential equations and provide numerical simulations for each of the mentioned results.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Furthermore, we employ the backward differentiation formula to solve the stiff differential equations and provide numerical simulations for each of the mentioned results.
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}
}
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}
}
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}
}