In the study of dynamical systems, there are many empirical rules that are valid for most systems people (in particular more applied ones) usually consider, but for which pathologic counterexamples exist.

For example, for most deterministic dynamics $X$, the following are equivalent:

• $X$ fulfils some definition of chaos.
• $X$ fulfils another definition of chaos.
• $X$ is bounded and the Lyapunov exponent of $X$ is positive.
• $X$ has a strange/fractal attractor (fractal dimension larger than topological dimension).
• $X$ passes any of the other empirical tests for chaos.

Yet there are pathological counterexamples to many of the equivalences, for example strange non-chaotic attractors. Being on the applied side, I cannot say much about how “typical” the counterexamples are and whether “simple” explanations exist except that we probably would not have differing definitions of chaos otherwise.