We are concerned with the possibility of modeling a given dynamical system by patching different dynamics, simpler than the given one. One interesting case is when the simpler dynamics correspond to integrable systems. We shall discuss the results of a recent work (Cavallari, Gronchi, Baù, Physica D 2022) about the case of the Sun-shadow dynamics, i.e. the planar motion of a mass particle in a force field defined by patching Kepler's and Stark's dynamics: this can be seen as a basic model for the motion of an Earth satellite perturbed by the solar radiation pressure and considering the Earth shadow effect.
The existence of periodic orbits of brake type is proved, and the Sun-shadow dynamics is investigated by means of a Poincaré-like map defined by a quantity that is not conserved along the flow. We also present the results of our numerical investigations on some properties of the map. Moreover, we construct the invariant manifolds of the hyperbolic fixed points related to the periodic orbits of brake type. The global picture of the map shows evidence of regular and chaotic behaviour.
Alain Albouy, Alain Chenciner, Jacques Laskar
