Abstract
In this work we propose the chaotic nonlinear Lü structure in quaternion form. Our new structure is high dimensional (9D). Highlights and qualities of this model are studied. Also, we present a novel king of quaternion synchronization. We call this new sort of synchronization “quaternion complete synchronization” (QCS). The idea of QCS did not consider in writing and has some unusual properties. Some organizations are conducted using the Lyapunov function to study QCS of two identical Lü attractors with quaternion variables. Numerical studies have shown the validity and accuracy of the analytical results of the control functions. Numerical studies also showed properties that distinguish QCS from the rest of the synchronization types that were previously studied.