1993/05/13 by Gert Eilenberger, G. Eilenberger, K. Schmidt +3
Computer Science · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #chao-dyn #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.chao-dyn/9305004
32 pages, TeX, preprint.sty file included
arxiv created 1993/05/13 · openalex publication_date 1993/05/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Bifurcation diagrams and plots of Lyapunov exponents in the r--Ω --plane for Duffing--type oscillators x +2r x +V'(x,Ωt) =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 Poincaré map of the system in action--angle variables. This map approaches the \it one dimensional form φn+1 = A + C \e-r T cos φn, T= π/ Ω provided \e-r T (but not necessarily C \e- r T), r and Ω are small. We derive asymptotic (for r, Ω 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'' ≥ 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 Poincaré map in powers of \e-rT. To first order, we obtain the φ--map, which survives asymptotically. To \it second order we obtain the two--dimensional I--φ--map. In I--direction it contracts by a factor \e-rT upon each iteration.