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We've all looked at stability domains for ODE time-steppers. At the most basic level, these are found by studying how the time-stepper handles the ODE x' = sigma x where sigma is a complex number with negative real part. This leads to a stability domain that has a continuous boundary. The underlying analysis generalizes to systems of ODEs if the linearized system is diagonalizable. In this talk, I'll discuss an implicit-explicit time-stepping scheme for which the linearized system is not diagonalizable; standard stability theory doesn't apply. I'll demonstrate that an adaptive time-stepper can be used to explore the stability domain and I'll give an example of a system for which the stability domain can have a discontinuous boundary; a small change in a parameter can lead to a jump in the stability threshold of the time-step size. This is joint work with my former PhD student, Dave Yan.