Dynamics of the Lorenz–Robbins system with control

In this paper, the existence of periodic orbits and homoclinic orbits in the Lorenz equations with high Rayleigh number r , i.e., the Lorenz–Robbins system, is rigorously proved by the generalized Melnikov method for the three-dimensional slowly varying systems. We analyze stability of these periodic orbits and show that the existence of these nontransverse but symmetrical homoclinic orbits implies the existence of chaos in the Lorenz–Robbins system. The results obtained analytically show the existence of chaotic dynamics in the Lorenz–Robbins system for the first time, but also solve a disagreement on the conditions of existence of periodic orbits in the system. In addition, a simple adaptive algorithm, which was recently developed by the author [D. Huang, Stabilizing near-nonhyperbolic chaotic systems with applications, Phys. Rev. Lett. 93 (2004) 214101] for stabilizing the near-nonhyperbolic chaotic systems, is used to successfully control the chaotic mixing of the Lorenz flows with high Rayleigh number found.