Resonant dynamics in strongly nonlinear systems

This work considers the effect of resonances in systems in which the two resonant frequencies are allowed to slowly change, depending on the state of the system. A strongly nonlinear system is introduced that allows for exact solutions. This system is then coupled to a second component, and through the method of averaging, a reduced-order model is developed that approximates the dynamical behavior near a 1:1 resonance between the two components. The resulting reduced system is studied using bifurcation theory and Melnikov analysis to obtain predictions of the near-resonant dynamics. Finally, these predictions are compared to numerical simulations of the original equations. Two main points appear: (i) for nonlinear systems, the period–amplitude dependence plays an important role in the evolution of the system, and (ii) the coordinates identified within the reduced system allow for the qualitative structure of the original equations to appear.