Foley, M. & Hall, D. (2014). MyGrammarLab Intermediate B1/B2 With Key. Harlow, United Kingdom: Pearson Longman
1 Participle clauses
Participle clauses give more information about a noun. We use the -ed or –ing form of the verb. Participle clauses don’t have a subject because their subject is the noun/pronoun in the main clause:
participle clause subject
Written in 1961, ‘Catch-22’ is a novel about a young American soldier, John Yossarian.
subject participle clause
In the middle of World War Two, he is sent to fight in Italy, leaving his friends and family behind.
FORMALITY CHECK
In written English we often put participle clauses in front of the main clause. We use a comma to separate the two clauses. Participle clauses are common in written English because they let us give a lot of information in a single sentence.
Participle clauses of reason, result, time, etc.
reason
Because he’s a student he can get a discount on rail travel.
Being a student, he can get a discount on rail travel.
result
A snowstorm covered the motorway. (The result was that dozens of drivers were trapped in their cars.)
A snowstorm covered the motorway, trapping dozens of drivers in their cars.
condition
If you give it enough water and sunlight, the plant will grow to three metres.
Given enough water and sunlight, the plant will grow to three metres.
time/ sequence
As I walked into the room I noticed the flowers by the window.
Walking into the room, I noticed the flowers by the window.
There is also a perfect form, having + past participle, which we can use to talk about an action that happened earlier:
Having paid the entrance fee, we walked into the museum.
(= After we had paid the entrance fee, we walked into the museum.)
Forms for active and passive meanings.
In participle clauses the -ing form has an active meaning:
The bank manager opened the safe and noticed something strange inside. (active verb)
➞ Opening the safe, the bank manager noticed something strange inside.
The -ed form has a passive meaning:
The flood victim stood on the roof. He was trapped by the rising water. (passive verb)
➞ The flood victim stood on the roof, trapped by the rising water.
MY COMMENTARY
I think we might be mixing up several concepts. I will analyse your examples from a relative clause point of view; for another point of view this won't apply and you'll have to read sumelic answer.
FOCUSING ON REDUCED RELATIVE CLAUSES
To be able to do this reduction, the subject of the reduced relative clause and the main sentence must be the same.
The key steps are:
+ identifying active or passive meaning;
+ identifying the order of the actions;
+ identifying the subject;
+ identifying whether we're talking about a reason, a cause, a condition or a time or sequence.
In example (1):
Which meaning has the verb we’re talking about? ACTIVE
+ Does he first escape from the NY tumult and then he goes on to do other things? If the answer is yes, then you need to include HAVING.
I wouldn't go for the choice of words in the example (1). I would have written a reduced relative clause.
Let's remind ourselves that we do not have to mix that up with the action having occurred before another action (in this case we would have to use a *perfect* form of the reduced relative clause, that is, HAVING + PAST PARTICIPLE, which, by the way, can only be used with active meaning verbs).
In (2) we cannot apply the reduced relative clause and you'll have to take it as an adjective, as sumelic suggests.
As for (3), we cannot omit having in (3) because he leaves the political life and then he goes on private, to advise Capital Group
(3') *Now, left political life, he advises Capital Group, the world's largest active fund manager, on important geopolitical issues.
In the event that you wouldn’t like to use a perfect form, and go directly with a –ED (past participle) or –ING (gerund participle) , you’d have to think about whether the reduced relative clause is giving a reason, a result, a time/sequence or a condition.
Also, you’ll have to think whether the meaning of the verb is active or passive.
(3') He began to advice Capital Group, the world's largest active fund manager, on important geopolitical issues, leaving his former political party in shambles.
(3') Backed properly, the Capital Group could become the world's largest active fund manager on important geopolitical issues.
The second example is a condition and the verb has a PASSIVE meaning, that’s why we need a PAST participle.