POINCARE MAPS OF DUFFING-TYPE OSCILLATORS AND THEIR REDUCTION TO CIRCLE MAPS .1. ANALYTIC RESULTS

Bifurcation diagrams and plots of Lyapunov exponents in the r-OMEGA-plane for Duffing-type oscillators x + 2rx + V'(x, OMEGAt) = 0 exhibit a regular pattern of repeating self-similar 'tongues' with complex internal structure. We demonstrate here that this behaviour is easily understood qualitatively and quantitatively from the Poincare map of the system in action-angle variables. This map approaches the one-dimensional form phi(n+1) = A + Ce(-rT) cos phi(n) T = pi/OMEGA provided e(-rT) (but not necessarily Ce(-rT)), r and OMEGA are small. We derive asymptotic (for small r, OMEGA) formulae for A and C for a special class of potentials V. We argue that these special cases contain all the information needed to treat the general case of potentials which obey V'' greater-than-or-equal-to 0 at all times. The essential tools of the derivation are the use of action-angle variables, the adiabatic approximation and the introduction of a non-oscillating reference solution of Duffing's equation, with respect to which the action-angle variables have to be determined. These allow the explicit construction of the Poincare map in powers of e(-r)T. To first order, we obtain the phi-map, which survives asymptotically. To second order we obtain the two-dimensional I-phi-map. In I direction it contracts by a factor e(-rT) upon each iteration.