We reanalyze the globally neutral non-Abelian black holes and present a unified picture, classifying them into two types: type I (black holes with a massless non-Abelian field) and type II (black holes with a "massive" non-Abelian field). For type II, there are two branches: The black hole in the high-entropy branch is "stable" and almost neutral, while that in the low-entropy branch, which is similar to the type I, is unstable and locally charged. To analyze their stabilities, we adopt the catastrophe-theoretic method, which reveals to us a universal picture of the stability of black holes. It is shown that the isolated type II black hole has a fold catastrophe structure. In a heat bath system, the type I black hole shows a cusp catastrophe, while the type II has both fold and cusp catastrophe.
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