Analysis of homoclinic bifurcation in Duffing oscillator under two-frequency excitation: Peculiarity of using Melnikov method in combination with averaging technique

We study Melnikov conditions predicting appearance of chaos in Duffing oscillator with hardening type of non-linearity under two-frequency excitation acting in the vicinity of the principal resonance. Since Hamiltonian part of the system contains no saddle points, Melnikov method cannot be applied directly. After separating the external force into two parts, we use a perturbation analysis that allows recasting the original system to the form suitable for Melnikov analysis. At the initial step, we perform averaging at one of the frequencies of the external force. The averaged equations are then analyzed by traditional Melnikov approach, considering the second frequency component of the external force and the dissipation term as perturbations. The numerical study of the conditions for homoclinic bifurcation found by Melnikov theory is performed by varying the control parameters of amplitudes and frequencies of the harmonic components of the external force. The predictions from Melnikov theory have been further verified numerically by integrating the governing differential equations and finding areas of chaotic behavior. Mismatch between the results of theoretical analysis and numerical experiment is discussed.