Open Access
Nonlinear stability of elliptic equilibria in Hamiltonian systems with exponential time estimates
(American Institute of Mathematical Sciences (AIMS), 2021) Cárcamo Díaz, Daniela Jacqueline; Palacián Subiela, Jesús Francisco; Vidal Díaz, Claudio; Yanguas Sayas, Patricia; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with n degrees of freedom we provide a criterion to obtain a type of formal stability, called Lie stability. Our result generalises previous approaches, as exponential stability in the sense of Nekhoroshev (excepting a few situations) and other classical results on formal stability of equilibria. In case of Lie stable systems we bound the solutions near the equilibrium over exponentially long times. Some examples are provided to illustrate our main contributions.
Open Access
Dynamics of axially symmetric perturbed Hamiltonians in 1:1:1 resonance
(Springer, 2018) Carrasco, Dante; Palacián Subiela, Jesús Francisco; Vidal Díaz, Claudio; Vidarte, Jhon; Yanguas Sayas, Patricia; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática
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.
Open Access
On the nonlinear stability of the triangular points in the circular spatial restricted three-body problem
(Pleiades Publishing, 2020) Cárcamo Díaz, Daniela Jacqueline; Palacián Subiela, Jesús Francisco; Vidal Díaz, Claudio; Yanguas Sayas, Patricia; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
The well-known problem of the nonlinear stability of L4 and L5 in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Moreover, in many cases we improve the estimates found in the literature.
Open Access
Dynamics in the charged restricted circular three-body problem
(Springer US, 2018) Palacián Subiela, Jesús Francisco; Vidal Díaz, Claudio; Vidarte, Jhon; Yanguas Sayas, Patricia; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
The existence and stability of periodic solutions for different types of perturbations associated to the Charged Restricted Circular Three Body Problem (shortly, CHRCTBP) is tackled using reduction and averaging theories as well as the technique of continuation of Poincaré for the study of symmetric periodic solutions. The determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the occurrence of Hamiltonian-Hopf bifurcations associated to some equilibrium points of the CHRCTBP.
Open Access
Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential
(IOP Publishing, 2019) Palacián Subiela, Jesús Francisco; Vidal Díaz, Claudio; Vidarte, Jhon; Yanguas Sayas, Patricia; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
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.
Open Access
Nonlinear stability in the spatial attitude motion of a satellite in a circular orbit
(Society for Industrial and Applied Mathematics Publications, 2021) Cárcamo Díaz, Daniela Jacqueline; Palacián Subiela, Jesús Francisco; Vidal Díaz, Claudio; Yanguas Sayas, Patricia; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
The paper considers the attitude nonlinear stability analysis of the spatial satellite problem and takes it one step further. A study of the Lie (formal) stability is presented and long-time estimates related to the Lie stable cases are provided. The connection with Nekhoroshev theory is also shown. Finally, KAM tori related to Lie stable, as well as unstable equilibria, are also calculated.