Cycle in a system close to a resonance system

An autonomous system when the frequencies of the linear system satisfy a relation which is close, with a frequency detuning ε, to an exact two-frequency internal resonance is investigated. The fact that there is an isolated periodic solution in the general situation is established. The cycle occurs at a distance O(ε) from zero in the case of third-order resonance and, at a distance of O(√ε) in the case of second- and fourth-order resonances. Systems of a general from as well as Lyapunov systems are considered. The problem of the stability of the cycles is studied. It is shown that, when small periodic perturbations of the order of μ act on the system being investigated, periodic motions exist in the μ χ neighbourhood of zero for which ε = O( μ k ). In addition, χ= 1 2 ,μ ⩾ 1 2 in the case of third-order resonance and χ = 1 3 , k ⩾ 2 3 in the case of second- and fourth-orderresonances.