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In this paper, we study the dynamics of the mappings θ 1 = θ + 2απ + μ 1 (θ) + o(r -1 ), r 1 = r + μ 2 (θ) + o(1), r → +∞, where a is a irrational rotation number. We prove the existence of orbits that go to infinity in the future or in the past by using the well-known Birkhoff Ergodic Theorem. Applying this conclusion, we deal with the unboundedness of solutions of Lienard equations with asymmetric nonlinearities.

Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities