Is there any systematic (algorithmic) method to convert a regular expression like
to an automaton (let's say a Buchi automaton)?
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There are (at least) three algorithms that convert a regular expression to an equivalent finite-state automaton:
Büchi automata enter the picture if you are working with $\omega$-regular expressions. See e.g. this description of the algorithm for constructing an equivalent Büchi automaton from an $\omega$-regular expression.