Periodic orbits and homoclinic orbits of the diffusionless Lorenz equations

In this Letter, the diffusionless Lorenz equations, which physically correspond to diffusionless convection, are studied qualitatively. Under the strong forcing, the model is reduced to a special case of slowly varying oscillators. Then the existence of three periodic orbits and two homoclinic orbits is proved rigorously by the Melnikov method. It is also shown that one of these periodic orbits is stable, and the other two fully unstable.