This paper investigates fast flavor conversions (FFCs) in a two-beam neutrino model, focusing on how the system evolves toward instability. The authors propose a model where the system is driven toward instability rather than starting in an already unstable state. They show that the final outcome depends on how the system is driven, and that the system generally sticks to the closest linearly stable configuration. This conclusion is proven using quasi-linear theory. The study considers two neutrino beams moving in opposite directions with different flavor content. The system is initially stable, but as it is driven toward instability, flavor conversions occur. The results show that the evolution is simpler to understand within the framework of quasi-linear theory, where small transverse oscillations are treated linearly, but their nonlinear feedback on the space-averaged configuration is accounted for. If driven slowly, the system tends to stick to the closest stable configuration along its entire evolution. The paper compares two scenarios: one where the beams appear suddenly (sudden case) and one where they are built up slowly. In the sudden case, the beams quickly exchange flavor, and the system oscillates without apparent damping. In the slow case, the system evolves more gradually, and the final outcome can differ significantly from the sudden case. The results show that the final state depends on the history of how the instability is realized. The authors also show that the system's final state can be influenced by the crossing of a configuration with zero polarization. This finding has implications for astrophysical settings, where the focus should be on the external dynamics that lead to the formation of the unstable state rather than on flavor instabilities in the neutrino radiation field. The paper concludes that the system always evolves toward the closest linearly stable configuration, a result that is supported by quasi-linear theory. This conclusion is consistent with previous findings that flavor conversions tend to remove angular crossings. The study provides an analytical explanation of this phenomenon using quasi-linear theory, which is typically applied to plasma systems. The results have implications for understanding the evolution of systems close to stability in astrophysical settings.This paper investigates fast flavor conversions (FFCs) in a two-beam neutrino model, focusing on how the system evolves toward instability. The authors propose a model where the system is driven toward instability rather than starting in an already unstable state. They show that the final outcome depends on how the system is driven, and that the system generally sticks to the closest linearly stable configuration. This conclusion is proven using quasi-linear theory. The study considers two neutrino beams moving in opposite directions with different flavor content. The system is initially stable, but as it is driven toward instability, flavor conversions occur. The results show that the evolution is simpler to understand within the framework of quasi-linear theory, where small transverse oscillations are treated linearly, but their nonlinear feedback on the space-averaged configuration is accounted for. If driven slowly, the system tends to stick to the closest stable configuration along its entire evolution. The paper compares two scenarios: one where the beams appear suddenly (sudden case) and one where they are built up slowly. In the sudden case, the beams quickly exchange flavor, and the system oscillates without apparent damping. In the slow case, the system evolves more gradually, and the final outcome can differ significantly from the sudden case. The results show that the final state depends on the history of how the instability is realized. The authors also show that the system's final state can be influenced by the crossing of a configuration with zero polarization. This finding has implications for astrophysical settings, where the focus should be on the external dynamics that lead to the formation of the unstable state rather than on flavor instabilities in the neutrino radiation field. The paper concludes that the system always evolves toward the closest linearly stable configuration, a result that is supported by quasi-linear theory. This conclusion is consistent with previous findings that flavor conversions tend to remove angular crossings. The study provides an analytical explanation of this phenomenon using quasi-linear theory, which is typically applied to plasma systems. The results have implications for understanding the evolution of systems close to stability in astrophysical settings.