‘Begun’ is used when talking about something that started in the past and is important now. It is the past participle form of 'begin'.
She has begun her homework now.
The show has just begun.
‘Begun’ is used with ‘have’ or ‘has’ to show an action started in the past and is still important or true now. This is called the present perfect tense.
They have begun working on the project.
We have begun to understand the problem.
In legal language, ‘begun’ means that an official process or document has started, often with formal importance.
The contract was begun on July 1st.
The investigation has begun according to law.
In expressions like ‘trouble has begun,’ ‘begun’ shows that a situation or process started and may continue with effects.
The trouble has just begun.
Once the discussion has begun, it is hard to stop.
