Computer-assisted proofs of existence of KAM tori in planetary dynamical models of \(\upsilon\)-And \(\mathbf{b}\). (English) Zbl 1532.70022
Summary: We reconsider the problem of the orbital dynamics of the innermost exoplanet of the \(\upsilon\)-Andromedæsystem (i.e., \(\upsilon\)-And \(\mathbf{b}\)) into the framework of a Secular Quasi-Periodic Restricted Hamiltonian model. This means that we preassign the orbits of the planets that are expected to be the biggest ones in that extrasolar system (namely, \(\upsilon\)-And \(\mathbf{c}\) and \(\upsilon\)-And \(\mathbf{d})\). The Fourier decompositions of their secular motions are injected in the equations describing the orbital dynamics of \(\upsilon\)-And \(\mathbf{b}\) under the gravitational effects exerted by those two exoplanets. By a computer-assisted procedure, we prove the existence of KAM tori corresponding to orbital motions that we consider to be very robust configurations, according to the analysis and the numerical explorations made in our previous article. The computer-assisted assisted proofs are successfully performed for two variants of the Secular Quasi-Periodic Restricted Hamiltonian model, which differs for what concerns the effects of the relativistic corrections on the orbital motion of \(\upsilon\)-And \(\mathbf{b}\), depending on whether they are considered or not.
MSC:
| 70H08 | Nearly integrable Hamiltonian systems, KAM theory |
| 70F15 | Celestial mechanics |
| 70H09 | Perturbation theories for problems in Hamiltonian and Lagrangian mechanics |
| 70H12 | Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics |
| 70-08 | Computational methods for problems pertaining to mechanics of particles and systems |
| 68V05 | Computer assisted proofs of proofs-by-exhaustion type |
Keywords:
normal form; Hamiltonian perturbation theory; exoplanet; secular quasi-periodic restricted Hamiltonian model; Fourier decompositionSoftware:
CAPD DynSysReferences:
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