The paper investigates the problem of the existence of periodic solutions for nonlinear Lienard-type differential equations with impulse action at non-fixed moments of time. Using the Poincare map method, the problem of finding periodic solutions is reduced to the analysis of the iterations of a one-dimensional map of an interval into itself. By means of mathematical induction, a theorem on the analytical form of periodic points of any natural period n is proved, and a strict structure of the corresponding polynomials with coefficients $±1$ is established. The conditions on the system parameters under which cycles corresponding to $T(n)$-periodic solutions exist are determined, and it is shown that these points obey Sharkovsky's order.

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