Arnold Flames and Resonance Surface Folds
by
Bruce B. Peckham
University of Minnesota
Coauthors: Richard P. McGehee

Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. The primary features of the resulting parameter plane bifurcation diagrams are "(Arnold) resonance horns" emanating from zero forcing amplitude. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper we investigate divisions of these horns into subregions - each subregion corresponding to maps having a constant number of periodic orbits. Subregions having more than the üsual" two periodic orbits can be interpreted as "folds" in the corresponding surface of fixed points in the phase × parameter space. Some of the resulting bifurcation pictures in the parameter plane appear in shapes we call Ärnold flames". This study leads to a simple method for constructing families of maps with bifurcation features such as flames and swallowtails. Results apply both to circle maps and forced oscillator maps.

Date received: February 1, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaa-58.