A hyperbolic twist mapping model for the study of asteroid orbits near the 3?1 resonance

Abstract

An algebraic mapping is found which contains all the properties of the dynamical system which describes the motion of an asteroid in the Sun-Jupiter system, in the restricted circular 3-body problem, near the 3∶1 resonance. The mapping is constructed by starting from the corresponding mapping of the two body problem, Sun-asteroid, in a rotating frame (twist mapping) and then by including the main effect of Jupiter, which is obtained analytically to be the generation of instability (hyperbolic mapping). The product of these two mappings is a hyperbolic twist mapping which gives invariant curves close to the actual case. This latter mapping depends on the energy, as a parameter. The properties of this mapping are studied, in connection with the evolution of an asteroid orbit which passes through the 3∶1 resonance due to a particular mode of energy dissipation.