Period-doubling cascades and devil’s staircases of the driven van der Pol oscillator

Bifurcation diagrams of the driven van der Pol oscillator are given showing mode-locking and period-doubling cascades. At low driving amplitudes locking regions occur following Farey sequences. At high driving amplitudes this relationship is destroyed due to the appearance of period-doubling cascades and coexisting attractors. A generalization of the winding number is used to compute devil’s staircases and winding-number diagrams of period-doubling cascades. The winding numbers at the period-doubling bifurcation points constitute an alternating sequence that converges at the accumulation point of the cascade.