An existence theorem of homoclinic orbit is given for second order Hamiltonian system -\"{x}(t) + a(t)x(t) - W_x(t, x(t)) = \lambda x(t) when \lambda=\lambda_1, where \lambda_1 is the first eigenvalue of operator Lx = \"{x} - a(t)x, and W_x(t, x) is sublinear growth in x \in R^n. When W_x(t, x) is odd in x, infinitely many distinct pairs of homoclinic orbits are obtained and the bifurcations occur for each \lambda\leq \lambda_1.

An existence theorem of homoclinic orbit is given for second order Hamiltonian system -\"{x}(t) + a(t)x(t) - W_x(t, x(t)) = \lambda x(t) when \lambda=\lambda_1, where \lambda_1 is the first eigenvalue of operator Lx = \"{x} - a(t)x, and W_x(t, x) is sublinear growth in x \in R^n. When W_x(t, x) is odd in x, infinitely many distinct pairs of homoclinic orbits are obtained and the bifurcations occur for each \lambda\leq \lambda_1.