The critical case of fourth-order resonance in a hamiltonian system with one degree of freedom

Abstract

The motion of a time-periodic Hamiltonian system with one degree of freedom in the neighbourhood of an equilibrium position is studied. It is assumed that the equilibrium is stable in the first approximation and that fourth-order resonance is present. The critical case is considered, when the system parameters are such that, in order to draw rigorous conclusions about the stability of the equilibrium, terms of order higher than four in the series expansion of the Hamiltonian must be taken into account. Sufficient conditions are derived for stability and instability, and the bifurcations of periodic motions are investigated in the neighbourhood of the equilibrium posiition when the system parameters pass through values corresponding to the critical case. The results are applied in the problem of the motion of a sphere in a uniform gravity field when there are collisions with the surface of an elliptic cylinder with a horizontal generator.