This paper is concerned with homoclinic orbits in the Hamiltonian system
(z) over dot = JH(z) (t, z),
where H is periodic in t with H-z (t, z) = L (t) z + R-z (t, z), R-z (t, z) = o(\z\) as z --> 0. We find a condition on the matrix valued function L to describe the spectrum of operator -(Td/dt + L) so that a proper variational formulation is presented. Supposing R-z is asymptotically linear as \z\ --> infinity and symmetric in z, we obtain infinitely many homoclinic orbits. We also treat the case where R-z is super linear as \z\ --> infinity with assumptions different from those studied previously in relative work, and prove existence and multiplicity of homoclinic orbits. Our arguments are based on some recent information on strongly indefinite functionals in critical point theory.