THE MOTION OF A NEARLY-HAMILTONIAN SYSTEM WITH ONE DEGREE OF FREEDOM IN THE CASE OF A FOURTH-ORDER RESONANCE

We consider the motion of a single-degree-of-freedom Hamiltonian system which is 2pi -periodic in time and which is subjected to the action of small dissipative forces defined by Rayleighʹs function. We assume that (i) the origin of coordinates is an equilibrium of the system, (ii) the linearized undamped system is stable, (iii) the characteristic exponents of the linearized system, ±iv, are pure imaginary, and (iv) the quantity 4v is close to an integer. The qualitative behavior of the approximate (model) system in a sufficiently small neighborhood of the origin is studied depending on the system parameters-the amount off resonance and the dissipation parameter. The existence, number, and stability of 8pi-periodic motions of the original system are studied rigorously.