Sign Stability via Root Locus Analysis
For researchers in network science, ecology, and economics who need stability guarantees from sign patterns alone, this work offers an intuitive understanding of existing opaque conditions, though it is incremental as it does not provide new necessary and sufficient conditions.
The paper uses root locus analysis to derive necessary conditions for sign stability (qualitative stability) in dynamical systems, where stability must hold for all parameter values consistent with given sign patterns. The approach provides a graphical interpretation of the known algebraic conditions.
With the rise of network science old topics in ecology and economics are resurfacing. One such topic is structural stability (often referred to as qualitative stability or sign stability). A system is deemed structurally stable if the system remains stable for all possible parameter variations so long as the parameters do not change sign. This type of stability analysis is appealing when studying real systems as the underlying stability result only requires the scientist or engineer to know the sign of the parameters in the model and not the specific values. The necessary and sufficient conditions for qualitative stability however are opaque. In order to shed light on those conditions root locus analysis is employed. This technique allows us to illustrate the necessary conditions for qualitative stability.