Non-linear oscillations of a Hamiltonian system with one degree of freedom and fourth-order resonance

Non-linear oscillations of a 2 pi-periodic Hamiltonian system with one degree of freedom are considered. It is assumed that the origin of coordinates is an equilibrium position, the linearized system is assumed to be stable, its characteristic exponents +/-iv are pure imaginary, and the value of 4v is close to an integer. When the methods of classical perturbation theory are used, the investigation reduces to an analysis of a model system which can be described by the typical Hamiltonian of problems on the motion of Hamiltonian systems with one degree of freedom in the case of fourth-order resonance. The system is analysed in detail. The results for the model system are applied to the total system using Poincare's theory of periodic motion and the KAM-theory. The existence, number and stability of 8 pi-periodic motions of the initial system are investigated. Trajectories of motion which start in a fairly small neighbourhood of the origin of coordinates are bounded. An estimate of the size of that neighbourhood is given. The examples considered are of a point mass above a curve in the shape of an ellipse which collides with the curve, and plane non-linear oscillations of a satellite in an elliptical orbit in the case of fourth-order resonance. (C) 1999 Elsevier Science Ltd. All rights reserved.