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For what it's worth - and not to distract from your real point - Exponential (or the more general Lyapunov) stability are a better explanation of instability. Stiffness is more a property of the method used to solve the ODE, rather than of the ODE itself.

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Stiffness is a property of certain dynamical systems, which certain ODE solvers are better suited for. That said, "stability" is an ugly word in numerical methods, particularly for differential equations, as it could mean a number of things. I think this is the source of the confusion here.

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