A quantitative stability and bifurcation analyses of the generalized duffing oscillator with strong nonlinearity

An analytical approach for the quantitative prediction of stability and bifurcations of approximate periodic solutions of the generalized Duffing oscillator ẍ + δẋ + α1x + α2x3 = L0 cos Ωt is presented. This approach is based on an improved harmonic balance method. Four types of Duffing oscillator are considered. Various stability and bifurcation conditions are derived. To verify the procedure, a comparison of the analytical results with numerical solutions is presented. Good agreement is observed. Numerical simulations using the analytically computed stability conditions are employed to expedite the numerical experimentations. Phase portraits for all four kinds of Duffing oscillator are shown. The number of stable and unstable solutions agree exactly with the analytical predictions.