Boundedness and synchronization of y-coupled Lorenz systems with or without controllers

The purpose of this paper is to investigate the boundedness and synchronization of y-coupled Lorenz systems. When the coupling term is only added to the second variable, we call them y-coupled Lorenz systems. In this paper, we first prove the boundedness of y-coupled Lorenz systems, which ensures the existence and uniqueness of the solution when t → ∞ . Based on the boundedness, we prove that for y-coupled Lorenz systems, QUAD condition is satisfied. It should be pointed out that QUAD condition plays an essential role in the discussion of synchronization. Based on the boundedness and QUAD condition, we prove that if the coupling is strong enough, the y-coupled Lorenz systems can achieve the complete synchronization globally and exponentially.