Poincaré maps of Duffing-type oscillators and their reduction to circle maps: II. Methods and numerical results

Bifurcation diagrams and plots of Lyapunov exponents in the r- plane for Duffing-type oscillators exhibit a regular pattern of repeating self-similar `tongues' with complex internal structure. We demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a Poincare map of the system in action-angle variables in the limit of large driving force or, equivalently, small driving frequency. This map approaches the one-dimensional form as derived in paper I. This second paper describes our approach to calculating the various constants and functions introduced in paper I. It gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of Duffing-type equations. Finally we show how to extend the range in the parameter space where the map is applicable.