K. Schmidt et G. Eilenberger, POINCARE MAPS OF DUFFING-TYPE OSCILLATORS AND THEIR REDUCTION TO CIRCLE MAPS - II - METHODS AND NUMERICAL RESULTS, Journal of physics. A, mathematical and general, 31(16), 1998, pp. 3903-3927
Bifurcation diagrams and plots of Lyapunov exponents in the r-Omega pl
ane for Duffing-type oscillators (x) over tilde + 2r (x) over dot + x(
q) = f(x, Omega t) exhibit a regular pattern of repeating self-similar
'tongues' with complex internal structure. We demonstrate here how th
is behaviour is easily understood qualitatively and quantitatively fro
m 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 phi(n+1) = alpha + beta cos p
hi(n) 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.