**FORMAS NORMALES EN SISTEMAS DINAMICOS HAMILTONIANOS: ASPECTOS TEORICOS Y APLICACIONES EN MECANICA CELESTE**

MTM2011-28227-C02-01

•

**Nombre agencia financiadora**Ministerio de Ciencia e Innovación

**Acrónimo agencia financiadora**MICINN

**Programa**Programa Nacional de Investigación Fundamental

**Subprograma**Investigación fundamental no-orientada

**Convocatoria**Investigación Fundamental No-Orientada

**Año convocatoria**2011

**Unidad de gestión**Sin informar

**Centro beneficiario**UNIVERSIDAD PÚBLICA DE NAVARRA (UPNA)

**Centro realización**DPTO. DE INGENIERIA MATEMATICA E INFORMATICA

**Identificador persistente**http://dx.doi.org/10.13039/501100004837

## Publicaciones

Found(s) 4 result(s)

Found(s) 1 page(s)

Found(s) 1 page(s)

#### Singular reduction of resonant Hamiltonians

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Meyer, Kenneth Ray
- Palacián Subiela, Jesús Francisco
- Yanguas Sayas, Patricia

This is an author-created, un-copyedited version of an article accepted for publication/published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/aab591., We investigate the dynamics of resonant Hamiltonians with n degrees of freedom to which we attach a small perturbation. Our study is based on the geometric interpretation of singular reduction theory. The flow of the Hamiltonian vector field is reconstructed from the cross sections corresponding to an approximation of this vector field in an energy surface. This approximate system is also built using normal forms and applying reduction theory obtaining the reduced Hamiltonian that is defined on the orbit space. Generically, the reduction is of singular character and we classify the singularities in the orbit space, getting three different types of singular points. A critical point of the reduced Hamiltonian corresponds to a family of periodic solutions in the full system whose characteristic multipliers are approximated accordingly to the nature of the critical point., The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and

Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain

and by the Charles Phelps Taft Foundation.

Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain

and by the Charles Phelps Taft Foundation.

#### Dynamics of axially symmetric perturbed Hamiltonians in 1:1:1 resonance

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Carrasco, Dante
- Palacián Subiela, Jesús Francisco
- Vidal Díaz, Claudio
- Vidarte, Jhon
- Yanguas Sayas, Patricia

This is a post-peer-review, pre-copyedit version of an article published in Journal of Nonlinear Science (2018) 28:1293–1359. The final authenticated version is available online at https://doi.org/10.1007/s00332-018-9449-y, We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by

normalisation up to terms of degree 4 in rectangular coordinates; after truncation of

higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system., The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and

Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of

Spain, and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of

Spain. D. Carrasco is also partially supported by Project DIUBB 165708 3/R, Universidad del Bío-Bío,

Chile and by FONDECYT Project 1181061, CONICYT (Chile).

normalisation up to terms of degree 4 in rectangular coordinates; after truncation of

higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system., The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and

Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of

Spain, and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of

Spain. D. Carrasco is also partially supported by Project DIUBB 165708 3/R, Universidad del Bío-Bío,

Chile and by FONDECYT Project 1181061, CONICYT (Chile).

#### Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Palacián Subiela, Jesús Francisco
- Vidal Díaz, Claudio
- Vidarte, Jhon
- Yanguas Sayas, Patricia

This is a peer-reviewed, un-copyedited version of an article published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/ab1bc6., A family of perturbed Hamiltonians H = 1/2 (x^2 + X^2) − 1/2 (y^2 + Y^2)+1/2

(z^2 + Z^2) + 2[ (x^4 + y^4 + z^4) + (x^2 y^2 + x^2 z^2 + y^2 z^2)] in 1: −1:1 resonance

depending on two real parameters is considered. We show the existence and

stability of periodic solutions using reduction and averaging. In fact, there are

at most thirteen families for every energy level h < 0 and at most twenty six

families for every h > 0. The different types of periodic solutions for every

nonzero energy level, as well as their bifurcations, are characterised in terms

of the parameters. The linear stability of each family of periodic solutions,

together with the determination of KAM 3-tori encasing some of the linearly

stable periodic solutions is proved. Critical Hamiltonian bifurcations on the

reduced space are characterised. We find important differences with respect

to the dynamics of the 1:1:1 resonance with the same perturbation as the one

given here. We end up with an intuitive interpretation of the results from a

cosmological viewpoint., The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain and MTM 2017-88137-C2-1-P of the Ministry of Science, Innovation and Universities of Spain. C Vidal is partially supported by Project Fondecyt 1180288.

(z^2 + Z^2) + 2[ (x^4 + y^4 + z^4) + (x^2 y^2 + x^2 z^2 + y^2 z^2)] in 1: −1:1 resonance

depending on two real parameters is considered. We show the existence and

stability of periodic solutions using reduction and averaging. In fact, there are

at most thirteen families for every energy level h < 0 and at most twenty six

families for every h > 0. The different types of periodic solutions for every

nonzero energy level, as well as their bifurcations, are characterised in terms

of the parameters. The linear stability of each family of periodic solutions,

together with the determination of KAM 3-tori encasing some of the linearly

stable periodic solutions is proved. Critical Hamiltonian bifurcations on the

reduced space are characterised. We find important differences with respect

to the dynamics of the 1:1:1 resonance with the same perturbation as the one

given here. We end up with an intuitive interpretation of the results from a

cosmological viewpoint., The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain and MTM 2017-88137-C2-1-P of the Ministry of Science, Innovation and Universities of Spain. C Vidal is partially supported by Project Fondecyt 1180288.

#### Invariant tori of rectilinear type in the spatial three-body problem

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Palacián Subiela, Jesús Francisco
- Sayas Bordonaba, Flora
- Yanguas Sayas, Patricia

In the context of the spatial three-body problem and KAM theory, specifically in the regime where the

Hamiltonian function is split as the sum of two Keplerian terms plus a small perturbation, we deal with

quasi-periodic motions of the three bodies such that two of the three particles (the so-called inner bodies)

describe near-collision orbits. More precisely the inner bodies never collide, but they follow orbits that

are bounded straight lines or close to straight lines. The motion of the inner bodies occurs either near the

axis that is perpendicular to the invariable plane (i.e. the fixed plane orthogonal to the angular momentum

vector that passes through the centre of mass of the system) or near the invariable plane. The outer particle’s

trajectory has an eccentricity varying between zero and a value that is upper bounded by eM

2 < 1 and lies

near the invariable plane. The three bodies’ orbits fill in invariant 5-tori and when the inner particles move

in an axis perpendicular to the invariable plane, they correspond to new solutions of the three-body problem.

Our approach consists in a combination of a regularisation procedure with the construction of various

reduced spaces and the explicit determination of sets of symplectic coordinates. The various reduced spaces

we build depend on what symmetries are taken into consideration for the reduction. Moreover we apply an

iso-energetic theorem by Han, Li and Yi on the persistence of quasi-periodic solutions for Hamiltonian

systems with high-order proper degeneracy. All these elements allow us to calculate explicitly the torsions

for the possible combinations that the three particles’ motions can achieve, The authors have received partial support by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain; MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain; and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of Spain.

Hamiltonian function is split as the sum of two Keplerian terms plus a small perturbation, we deal with

quasi-periodic motions of the three bodies such that two of the three particles (the so-called inner bodies)

describe near-collision orbits. More precisely the inner bodies never collide, but they follow orbits that

are bounded straight lines or close to straight lines. The motion of the inner bodies occurs either near the

axis that is perpendicular to the invariable plane (i.e. the fixed plane orthogonal to the angular momentum

vector that passes through the centre of mass of the system) or near the invariable plane. The outer particle’s

trajectory has an eccentricity varying between zero and a value that is upper bounded by eM

2 < 1 and lies

near the invariable plane. The three bodies’ orbits fill in invariant 5-tori and when the inner particles move

in an axis perpendicular to the invariable plane, they correspond to new solutions of the three-body problem.

Our approach consists in a combination of a regularisation procedure with the construction of various

reduced spaces and the explicit determination of sets of symplectic coordinates. The various reduced spaces

we build depend on what symmetries are taken into consideration for the reduction. Moreover we apply an

iso-energetic theorem by Han, Li and Yi on the persistence of quasi-periodic solutions for Hamiltonian

systems with high-order proper degeneracy. All these elements allow us to calculate explicitly the torsions

for the possible combinations that the three particles’ motions can achieve, The authors have received partial support by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain; MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain; and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of Spain.