Structure in the bifurcation diagram of the Duffing oscillator

Abstract

We identify four levels of structure in the bifurcation diagram of the two-well periodically driven Duffing oscillator, plotted as a function of increasing control parameter T, the period of the driving term. The superstructure, or bifurcation peninsula, repeats periodically as T increases by ∼2π, beginning and ending with symmetric period-one orbits whose local torsions differ by 2. Within each bifurcation peninsula there is a systematic window structure. The primary window structure is due to Newhouse and Newhouse-like orbits. Fine structure is due to a Farey sequence of well-ordered orbits between the primary windows. Hyperfine structure consists of very narrow windows associated with non-well-ordered orbits. We construct a template for the Duffing oscillator, a two-dimensional return map, and a one-dimensional return map which describes the systematics of orbit creation and annihilation. All structures are identified by topological indices. Our predictions are based on, and compatible with, numerical computations.