The double mathematical pendulum (DMP) is a Hamiltonian system. The homoclinic transversal intersections are studied. The system has two degrees of freedom. The system exhibits a regular and a chaotic behavior of its trajectories. Hyperbolic periodic trajectories in the stochastic component are obtained. The Poincare-Arnold-Melnikiov method is used to obtain the homoclinic points. Asymptotic formula for the homoclinic invariant of homoclinic trajectory is obtained.