This paper utilizes Cauchy's argument principle in the frequency domain to develop novel stability maps, providing guidelines for measures which can be taken to mitigate thermoacoustic instabilities in combustion appliances. The existing approaches mainly concentrate on identifying the onset of thermoacoustic instabilities by calculating unstable frequencies and growth rates. However, they provide limited practical guidance for modifying system characteristics, especially those dependent on frequency, to achieve flame stabilization. In the present contribution, several thermoacoustic stability criteria are introduced that leverage the Cauchy's argument principle and direct evaluation of the dispersion relation's argument. These criteria offer deeper insights and facilitate a systematic flame stabilization process by enabling modifications to both the (passive) acoustic subsystems and/or the (active) subsystem containing the combustion processes. This approach allows for a construction and comprehensive understanding of the stability map for a given thermoacoustic system, leading to more effective guidelines to elaborate and implement the combustion system stabilization strategies. To demonstrate the practical application of this framework, two illustrative thermoacoustic systems are discussed. Novelty and significance statement This study introduces a novel framework for assessing thermoacoustic stability.

• It provides a method for detecting the onset of thermoacoustic instability and offers valuable insights into critical frequency ranges. This approach facilitates the identification of necessary modifications in flame and acoustic subsystems across different frequencies to achieve system stabilization.

• This framework allows for the selection of the most suitable stability criterion based on the available combustion system's characteristics. For example, by knowing acoustic properties in both upstream and downstream components, either the DDS or DCS criterion can generate a comprehensive, frequency-dependent stability map for flame transfer function values. This approach eliminates the need for iterative integration, differential equations, or direct solutions to dispersion relations.