On the growth of the number of long periodic solutions of differential equations (Q1316813)

We show that the number of simple closed trajectories of period less than \(N\) for the differential equation defined by a vector field on a compact manifold of dimension greater than 2 may grow with \(N\) at an arbitrarily fast rate. Main theorem. Let \(M\) be a compact smooth manifold of dimension \(\geq 3\) and let \(A_ 1,A_ 2,\dots\) be a sequence of positive integers. Then there exists a smooth vector field on \(M\) with nondegenerate singular points such that all its closed trajectories are simple and the number of such trajectories of period not greater than \(N\) exceeds \(A_ N\) for some increasing sequence of positive integer values of \(N\).