The dynamical behavior of a 5-dimensional Lorenz model (5DLM) is investigated. Bifurcation diagrams address the chaotic and periodic behaviors associated with the bifurcation parameter. The Hamilton energy and its dependence on the stability of the dynamical system are presented. The global exponential attractive set (GEAS) is estimated in different 3-dimensional projection planes. A more conservative bound for the system is determined, that can be applied in synchronization and chaos control of dynamical systems. Finally, the finite time synchronization of the 5DLM, indicating the role of the ultimate bound for each variable, is studied. Simulations illustrate the effectiveness of the achieved theoretical results.