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Liquid argon detectors are employed in a wide variety of nuclear and particle physics experiments. The addition of small quantities of xenon to argon modifies its scintillation, ionization, and electroluminescence properties and can improve its performance as a detection medium. However, a liquid argon-xenon mixture can develop instabilities, especially in systems that require phase transitions or that utilize high xenon concentrations. In this work, we discuss the causes for such instabilities and describe a small (liter-scale) apparatus with a unique cryogenic circuit specifically designed to handle argon-xenon mixtures. The system is capable of condensing argon gas mixed with O(1%) xenon by volume and maintains a stable liquid mixture near the xenon saturation limit while actively circulating it in the gas phase. We also demonstrate control over instabilities that develop when the detector condition is allowed to deviate from optimized settings. This progress enables future liquid argon detectors to benefit from the effects of high concentrations of xenon doping, such as more efficient detection of low-energy ionization signals. This work also develops tools to study and mitigate instabilities in large argon detectors that use low concentration xenon doping.