When dimension n≥3, the existence, nonexistence, uniqueness and stability of periodic orbits of dynamical systems, in the theory and application, are worthy of study. in the two-dimensional dynamical systems, the basic tools of study the existence of periodic orbits is Poincar-Bendixson and its corollary (ring zone principle).However, The counter example of D. Heedene display that Poincar-Bendixson has not been simply popularized in the high dimensional case. Thus, over the years, studying the existence of periodic orbits in the dynamical systems, mainly rely on the ring zone principle. To the quasi-geostrophic model in the action of the three waves, the simple pattern of exogenous, turbulent friction has been more mature, but there is less study about it is cyclical behavior. Furthermore, homoclinic orbit and heteroclinic orbit are associated with solitary wave and the vortex, and homoclinic orbit and heteroclinic orbit at rail cross-sectional intersect is the condition of chaos, also the key to study the certainty and randomness. The combination of certainty and randomness is the main methods that quasi-geostrophic model explaining the turbulent structure. So, homoclinic orbit and heteroclinic orbit are very important to the periodic and stability of quasi-geostrophic model. Through analyzing, we discover that the nonlinear has a form of three-dimensional- Hamilton, induced by the main structure-Jacobi of the quasi-geostrophic model in the action of three-wave, so the article study the topological nature of the model based on Poincar-Bendixson. Meanwhile, in view of the importance of the quasi-geostrophic model for the structure of turbulence, homoclinic orbit and heteroclinic orbit that the balance point of the four saddle pattern corresponding to were analyzed. The analysis revealed that the presence of four different homoclinic (heteroclinic) orbit as well as the center for each type of balance exist in a column corresponding periodic orbits around the center on the Poisson symplectic leaf surface. Having a discussion about a special class of perturbations of the situation and the system of central periodic orbit perturbations near the equilibrium point, people are able to access to the conditions of the original three-wave quasi-geostrophic model of periodic solutions exist.