Short periodic orbits for the Lorenz system

The existence of short periodic orbits for the Lorenz system is studied rigorously. We describe a method for finding all short cycles embedded in the chaotic attractor. We use the method of close returns to find initial points for the Newton operator, combined with interval tools for proving the existence of periodic orbits in a neighborhood of a pseudo-periodic orbit. All periodic orbits with period p les 8 of the Poincare map for the Lorenz system are found.