Summary: We find new conditions, which are different from those used in previous related studies, to ensure the existence of infinitely many homoclinic orbits for the second order Hamiltonian systems of the form $-\ddot q = V_q (t,q).$ Here, we assume that $$V(t,q)$$ depends periodically on $$t$$, and assume, on $$q$$, that $$V(t,q)$$ is asymptotically quadratic at $$q=0$$ and is, as $$|q|\rightarrow \infty$$, either asymptotically quadratic or superquadratic, as well as the new conditions.