Poincar\'e 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 $$\ddot x +2r\dot x +V'(x,\Omega t) =0$$ 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 $$\varphi_{n+1} = A + C \e^{-r T} \cos \varphi_n, \ \ T= \pi / \Omega$$ provided $\e^{-r T}$ (but not necessarily $C \e^{- r T}$), $r$ and $\Omega$ are small. We derive asymptotic (for $r$, $\Omega$ 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 $\e^{-rT}$. To first order, we obtain the $\varphi$--map, which survives asymptotically. To {\it second} order we obtain the two--dimensional $I$--$\varphi$--map. In $I$--direction it contracts by a factor $\e^{-rT}$ upon each iteration.