Bifurcations of an oscillator to two-harmonic excitation

The incremental harmonic balance method has been successful for harmonic excitation. It is extended to determine the steady-state solutions of a non-linear oscillator subject to periodic (two-harmonic) excitation. Higher-order subharmonic solutions result from bifurcations. As the bifurcation process continues in an accelerated rate, chaotic solutions are obtained when no simple subharmonic solution coexists. When periodic solutions do coexist, the final steady-state solution depends on the initial conditions. The evolution of the amplitude against the system parameters can be recorded on a bifurcation graph. An initial bifurcation graph is constructed when one of the system parameters varies. Neighbouring bifurcation graphs when other system parameters are changing are obtained in an incremental manner. If only the boundaries dividing the qualitatively different solutions are constructed, a parametric diagram is obtained. The characteristic of the solutions can be read directly from the diagram. For an oscillator subject to two-harmonic excitation, the parametric diagram is found to be qualitatively different from those with one-harmonic excitation. The parametric diagram is highly foliated when many stable and unstable higher-order subharmonic solutions coexist at the same time under some combination of conditions. It is possible that the periodic solutions disappear suddenly and give way to chaotic solutions due to a small change in the system parameters without undergoing period doubling.