2005 PT Cruiser GT Conv.
This is difficult to answer without more data. That being said there are a number of things that would cause this to happen: engine running and running hot, when the thermostat opens, the car may surge a bit to cool it down; the fuel filter may be clogged or the pump may be malfunctioning. It it is the HO Turbo, it could be an issue with that. Run a diagnostic on it. Do any of the lights on the dash light up when this happens?
Try running some good gas through it like Chevron or get a fuel additive like Techron and add it to your next tank. I've had two GT's and this has helped. This would help the fuel and your intake system and would be the easiest thing to do.
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Kelly Little Ashford