5.
Pedro C. Pereira; Douglas D. Novaes; Murilo R. Cândido
A mechanism for detecting normally hyperbolic invariant tori in differential equations Journal Article
Em: Journal de Mathématiques Pures et Appliquées, vol. 177, não 1, pp. 45, 2023.
@article{PerNovCan23,
title = {A mechanism for detecting normally hyperbolic invariant tori in differential equations},
author = {Pedro C. Pereira and Douglas D. Novaes and Murilo R. Cândido},
url = {https://arxiv.org/abs/2208.10989},
doi = {10.1016/j.matpur.2023.06.008},
year = {2023},
date = {2023-09-01},
urldate = {2023-04-28},
journal = {Journal de Mathématiques Pures et Appliquées},
volume = {177},
number = {1},
pages = {45},
abstract = {Determining the existence of compact invariant manifolds is a central quest in the qualitative theory of differential equations. Singularities, periodic solutions, and invariant tori are examples of such invariant manifolds. A classical and useful result from the averaging theory relates the existence of isolated periodic solutions of non-autonomous periodic differential equations, given in a specific standard form, with the existence of simple singularities of the so-called guiding system, which is an autonomous differential equation given in terms of the average in time of the original non-autonomous one. In this paper, we provide an analogous result for the existence of invariant tori. Namely, we show that a non-autonomous periodic differential equation, given in the standard form, has a normally hyperbolic invariant torus in the extended phase space provided that the guiding system has a hyperbolic limit cycle. We apply this result to show the existence of normally hyperbolic invariant tori in a family of jerk differential equations.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Determining the existence of compact invariant manifolds is a central quest in the qualitative theory of differential equations. Singularities, periodic solutions, and invariant tori are examples of such invariant manifolds. A classical and useful result from the averaging theory relates the existence of isolated periodic solutions of non-autonomous periodic differential equations, given in a specific standard form, with the existence of simple singularities of the so-called guiding system, which is an autonomous differential equation given in terms of the average in time of the original non-autonomous one. In this paper, we provide an analogous result for the existence of invariant tori. Namely, we show that a non-autonomous periodic differential equation, given in the standard form, has a normally hyperbolic invariant torus in the extended phase space provided that the guiding system has a hyperbolic limit cycle. We apply this result to show the existence of normally hyperbolic invariant tori in a family of jerk differential equations.
4.
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}
}
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.
3.
Murilo R. Cândido; Douglas D. Novaes; Claudia Valls
Periodic solutions and invariant torus in the Rössler System Journal Article
Em: Nonlinearity, vol. 33, não 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},
urldate = {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.
2.
Murilo R. Cândido; Douglas D. Novaes
On the torus bifurcation in averaging theory Journal Article
Em: Journal of Differential Equations, vol. 268, não 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},
urldate = {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.
1.
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.
