THE ONSET OF BIFURCATIONS IN THE FORCED DUFFING EQUATION WITH DAMPING

Abstract

The paper presents a complete description of the periodic solutions of the Duffing equation: [Formula: see text] for large values of the forcing Γ and the damping Δ. It contains a proof that the equation admits of an infinite sequence of bifurcation curves in the Γ - Δ plane, alternately of the saddle-node and odd periodic — simply 2π-periodic type, whose maxima lie at large Γ along the line: [Formula: see text] where C(Γ) has a finite limit as Γ → ∞. The positions of the maxima are interlaced in asymptotically equal intervals of Γ1/3, with a spacing of 1.403 units. For Δ > Δc(Γ), the Duffing equation admits of a unique periodic solution if Γ is high enough. These results are obtained by showing that the half-period Poincaré map offered by the Duffing equation is asymptotically equivalent to a map of a circle into itself, according to: [Formula: see text] where χ is an angular variable and β, Σ depend on Γ, Δ. The numerical constants appearing in the circle map and its corrections are determined in the limit Γ → ∞ by a parameter-free boundary layer equation and its variation.