Abstract
A multi-time average variational principle is developed for a study of subharmonic bifurcations in a nonlinear ordinary differential equation, and an analytic method is established to calculate the values of the control parameter at the bifurcation points. A specific application is given to the analysis of the Duffing equation x ̈ + 2λ x ̇ + x − 4x 3 = ⨍ cos ωt . The connection between period-doubling bifurcations and parametric resonance is clarified. With ⨍ the control parameter, the critical values are evaluated analytically at a few bifurcation points with the results in excellent agreement with those of the numerical integration.
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