This paper explores the dynamics of $2$D spatiotemporal discrete systems, focusing on the stability and bifurcations of periodic solutions, particularly $3$-cycles. After introducing the concept of a third-order cycle, we discuss both numerical and analytical techniques used to analyze these cycles, defining four types of $3$-periodic points and their associated stability conditions. As a specific case, this study examines a spatiotemporal quadratic map, analyzing the existence of $3$-cycles and various bifurcation scenarios, such as fold and flip bifurcations, as well as chaotic behavior. In $2$D spatiotemporal systems, quadratic maps intrinsically offer better conditions that favor the emergence of chaos, which is characterized by high sensitivity to initial conditions. The findings emphasize the complexity of these systems and the crucial role of bifurcation curves in understanding stability regions. The paper concludes with key insights and suggestions for future research in this field.