Abstract We study a normal periodic system of ordinary differential equations with a small parameter, which is quasilinear in a neighborhood of infinity, under the assumption that the right-hand side of the system has a critical linear approximation. In terms of the properties of the first homogeneous nonlinear approximation of the monodromy operator, we obtain conditions for the existence of a periodic solution whose initial value is infinitely large for an infinitesimal value of the parameter.
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