A Nonlinear Oscillator with a Strange Attractor

P. Holmes


We study the nonlinear oscillator <latex>$\ddot{x}$</latex>+<latex>$\delta \dot{x}$</latex>-<latex>$\beta $</latex>x+<latex>$\alpha $</latex>x<latex>$^{3}$</latex> = f cos (<latex>$\omega $</latex>t) (A) from a qualitative viewpoint, concentrating on the bifurcational behaviour occurring as f <latex>$\geq $</latex> 0 increases for <latex>$\alpha $</latex>, <latex>$\beta $</latex>, <latex>$\delta $</latex>, <latex>$\omega $</latex> fixed > 0. In particular, we study the global nature of attracting motions arising as a result of bifurcations. We find that, for small and for large f, the behaviour is much as expected and that the conventional Krylov-Bogoliubov averaging theorem yields acceptable results. However, for a wide range of moderate f extremely complicated non-periodic motions arise. Such motions are called strange attractors or chaotic oscillations and have been detected in previous studies of autonomous o.d.es of dimension <latex>$\geq $</latex> 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 means of the Poincare map associated with (A). Using analogue and digital computer simulations, we provide a fairly complete characterization of the strange attractor arising for moderate f. This ergodic motion arises naturally from the deterministic differential equation (A).

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