Partial normal form near a saddle of a Hamiltonian system

For a smooth or real analytic Hamiltoniain vector field with two degrees of freedom we derive a local partial normal form of the vector field near a saddle equilibrium (two pairs of real eigenvalues $\pm \lambda_1$, $\pm \lambda_2$, $\lambda_1 > \lambda_2 > 0$). Only a resonance $\lambda_1 = n \lambda_2$ (if is present) influences on the normal form. This form allows one to get convenient almost linear estimates for solutions of the vector field using the Shilnikov's boundary value problem. Such technique is used when studying the orbit behavior near homoclinic orbits to saddle equilibria in a Hamiltonian system. The form obtained depends smoothly on parameters, if the vector field smoothly depends on parameters