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On the existence of limit cycles of the perturbed harmonic oscillator in the degenerate case

We investigate the existence of limit cycles of the equation x + x = e f(x,x), e small, assuming that the equations for their generating amplitudes have a multiple root. The method used is to represent the original problem in terms of the Lyapunov-Schmidt operator system and then study its solvability by means of iterating and applying the comparison theorem. For some important cases of f being a polynomial function of x only, the distribution of roots of the system of comparison is completely investigated by methods of algebraic simplification and constructing the generalized Sturm sequences. If several degenerate limit cycles correspond to a root of the equation for generating amplitudes of multiplicity n ≥ 2, they are separated by the distance being of order O(e) or higher, and there is a conjecture that the maximal number of them is n.