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The formation of patterns in driven systems has been studied extensively, and their emergence can be connected to a fine balance of instabilities and stabilization mechanisms. While the early phase of pattern formation can be understood on the basis of linear stability analyses, the long-time dynamics can only be described by accounting for the interactions between the excitations generated by the drive. Here, we observe the stabilization of square patterns in an interaction-driven, two-dimensional Bose-Einstein condensate. These patterns emerge due to inherent high-order processes that become relevant in the regime of large phonon occupations. Theoretically, this can be understood as the emergence of a stable fixed point of coupled nonlinear amplitude equations, which include phonon-phonon interactions. We experimentally probe the predicted flows towards such a stable fixed-point, as well as repulsion from a saddle fixed-point, using the experimental control unique to quantum gases.