Poincare maps of Duffing-type oscillators and their reduction to circle maps. I. Analytic results

Abstract

Bifurcation diagrams and plots of Lyapunov exponents in the r- Omega -plane for Duffing-type oscillators x+2rx+V'(x, Omega t)=0 exhibit a regular pattern of repeating self-similar 'tongues' with complex internal structure. The authors 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+C e-rT cos phi n;T= pi / Omega ; provided e-rT (but not necessarily Ce-rT), r and Omega are small. The authors derive asymptotic (for small r, Omega ) formulae for A and C for a special class of potentials V. They argue that these special cases contain all the information needed to treat the general case of potentials which obey V">or=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-rT. To first order, the authors obtain the phi -map, which survives asymptotically. To second order they obtain the two-dimensional I- phi -map. In the I direction it contracts by a factor e-rT upon each iteration.