Local Dynamics at Focal Points Coupled with Elliptic Directions

We investigate local dynamics of a 4-dimensional system with small and slowly varying time periodic forcing. By assuming the unperturbed system is autonomous and has a fixed point with eigenvalues $(0,0,i,-i)$ ( 0 , 0 , i , ? i ) , we study homoclinic, subharmonic solutions and Hopf bifurcation in a $O(epsilon )$ O ( ? ) neighborhood of the fixed point, where $epsilon $ ? is the perturbation parameter.