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We study the nonlinear oscillator ẍ + δẋ — βx + αx³ = f cos (ωt) (A) from a qualitative viewpoint, concentrating on the bifurcational behaviour occurring as f > 0 increases for α, β, δ, ω to fixed > 0. In particular, we study the global nature of attracting motions arising as a result of bifurcations. We find that, for small and for large f, the behaviour is much as expected and that the conventional Krylov-Bogoliubov averaging theorem yields acceptable results. However, for a wide range of moderate f extremely complicated non-periodic motions arise. Such motions are called strange attractors or chaotic oscillations and have been detected in previous studies of autonomous o.d.es of dimension > 3. In the present case they are intimately connected with homoclinic orbits arising as a result of global bifurcations. We use recent results of Mel’nikov and others to prove that such motions occur in (A) and we study their structure by …