Resonance and Symmetry Breaking for a Duffing Oscillator

The solution of $\ddot x + 2\delta \dot x + x( {1 - x^2 } ) = \varepsilon \sin \omega t$ is approximated by $x = a_0 + a_1 \sin \omega t$, and $a_1 = a_1 ( \omega ;\delta ,\varepsilon )$ is determined on the hypothesis that $a_0 = 0$. It is shown that this symmetric solution is stable, except on that segment of the resonance curve ($a_1$ vs $\omega $) that connects the turning points (if any), if $\varepsilon < \varepsilon _{\text{x}} = ( 2 /\sqrt{3})\delta ( 1 - \delta ^2 )$ and $a_1^2 < a_{\bf x}^2 = ( 2/3)( 1 - \delta ^2 )$. The resonance curve crosses itself at $a_1 = a_{\bf x} $ if $\varepsilon = \varepsilon _{\bf x} $, and the resonance curves for $\varepsilon > \varepsilon _{\bf x} $ have no maxima and consist of two separate branches. Symmetry breaking occurs at $a_1^2 = 2/3$ if $\varepsilon > \varepsilon _{\bf x} $, and the symmetric solution loses stability to an asymmetric solution (for which $| a_0 | > 0$) through a subcritical pitchfork bifurcation. An independent calculation, which is exact in the limit $\varepsilon \downarrow \infty $ with $\delta = O( \varepsilon )$ and yields results for symmetry breaking that are in agreement with those observed in numerical experiments, reveals that the error in the preceding approximations to $a_{\text{x}} $ and $\varepsilon _{\text{x}} $ is 3 percent. A comparison with Mel’nikov’s criterion for the transition to chaos is made. A formal expansion of the solution in Lamé functions is developed in an appendix. An extended Fourier-series calculation based on numerical collocation is reported by Bryant in a second appendix.