steering wheel shakes hard when turning right to left viseversa n cold weather.

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Steering wheel shakes hard when turning quickly right to left or left to right and seems to be more problematic in cold weather? the dashboard warning light ( ! ) comes on and then diagnostic panel failure box comes on and highlights the (power steering icon) as being the problem. Just B4 the warning lights turn on the steering gets stiff, especially as you slow down as if the Power steering pump has failed or the PS belt jumped track or broken. Then when you park, turn off the car, then restart it all is fine until you travel about 4 or 5 more miles it will repeat the same thing again. There are no noises during the described events, no broken hoses, apparent leaks anywhere the car has sit or been parked.