Melnikovʹs Method and Codimension-Two Bifurcations in Forced Oscillations

Using a Melnikov-type technique, we study codimension-two bifurcations called the Bogdanov–Takens bifurcations for subharmonics in periodic perturbations of planar Hamiltonian systems. We give a criterion for the occurrence of the Bogdanov–Takens bifurcations and present approximate expressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov–Takens bifurcation points. We illustrate the theoretical result with an example.