Poincar\'e maps of Duffing--type oscillators and their reduction to circle maps. I. Analytic results

Bifurcation diagrams and plots of Lyapunov exponents in the $r$–$\mathrm{\Omega }$ –plane for Duffing–type oscillators $$\ddot{x}+2r\dot{x}+V^{\mathrm{\prime }}(x,\mathrm{\Omega }t)=0$$ exhibit a regular pattern of repeating selfsimilar ``tongues’’ with complex internal structure. We demonstrate here that this behaviour is easily understood qualitatively and quantitatively from the Poincar'e map of the system in action–angle variables. This map approaches the {\it one dimensional} form $${\phi }_{n+1}=A+C{\text{e}}^{-rT}\cos {\phi }_n,T=\pi \mathrm{/}\mathrm{\Omega }$$ provided ${\text{e}}^{-rT}$ (but not necessarily $C{\text{e}}^{-rT}$), $r$ and $\mathrm{\Omega }$ are small. We derive asymptotic (for $r$, $\mathrm{\Omega }$ small) 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 $V’’ \ge 0$ at all times. The essential tools of the derivation are the use of action–angle variables, the adiabatic approximation and the introduction of a nonoscillating 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 Poincar'e map in powers of ${\text{e}}^{-rT}$. To first order, we obtain the $\phi$–map, which survives asymptotically. To {\it second} order we obtain the two–dimensional $I$–$\phi$–map. In $I$–direction it contracts by a factor ${\text{e}}^{-rT}$ upon each iteration.