The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects

We describe numerical tools for the stability analysis of extrasolar planetary systems. In particular, we consider the relative Poincare variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator MEGNO (a measure of the maximal Lyapunov exponent) that helps to distinguish chaotic and regular configurations. The results concerning the three-planet extrasolar system HD 37124 are presented and discussed. The best fit solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best fit solutions are found in dynamically active region of the phase space. The long term stability of the system is determined by a net of low-order two-body and three-body mean motion resonances. In particular, the three-body resonances may induce strong chaos that leads to self-destruction of the system after Myrs of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behavior.Comment: 11 pages (total), 8 figures. Accepted for publication in MNRAS. The definitive version will be/is available at http://www.blackwellpublishing.com. The astro-ph version is prepared with low resolution figures. To obtain the manuscript with full-resolution figures, please visit http://www.astri.uni.torun.pl/~chris/mnrasIII.ps.g

Poisson integrators

An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. Numerical integrators using generating functions, Hamiltonian splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems and the discrete gradient methods are also presented.Comment: 30 page

An introduction to Lie group integrators -- basics, new developments and applications

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups