Forced secondary resonances at the 3 : 1 commensurability

Abstract

For the 3 : 1 Jovian resonance problem, the time scales of the two degrees of freedom of the resonant Hamiltonian are well-separated [5]. With the adiabatic approximation, the solution for the fast oscillations can be found in terms of the slowly varying variables. Thus the rapidly oscillating terms in the slow oscillation equations can be treated as forced terms. We refer to the resonance between the forcing and intrinsic frequencies as a forced secondary one in this paper. We discuss the forced secondary resonances in asteroidal motion at the 3 : 1 commensurability by using Wisdom's method. The results show that the orbits situated originally near the resonance will leave the neighbourhood of resonance and tend to the separatrices and critical points for different energies, respectively. We have not found any stochastic web as expected in this case. Moreover, we study the problem of validity on the approximation of a system.