DA50 flight crew at FL370 reports TCAS event with two airliners at FL360 and FL380. DA50 crew receives RA to descend then climb triggering RA in at least one airliner and alerting ATC.

Narrative

2 airliners were involved. One at FL360 and one at FL380 and our Falcon 50 at FL370. At approximately XA00Z we received a traffic alert on our TCAS. Traffic was approaching from our 12 o'clock position and 1;000 FT above. We received an RA stating 'Monitor Vertical Speed' and a green indication to descend at 300 FPM. Very soon thereafter another traffic alert followed by 'Monitor Vertical Speed' and a command of 300 FPM climb. This traffic was also at our 12 o'clock position and below us. This all took place within seconds and was very disconcerting to both myself and the traffic below. I believe that we never gained or lost more than 200 FT. The human factors involved were the reception of 2 RAs in rapid succession in different directions; looking out the window for visual acquisition; disengaging the autopilot and following the commands of the TCAS in 2 different directions at high altitude where the flight controls are very sensitive. This can create a very tense situation. I think that although you must respond to the initial alert within 5 seconds; it is best not to rush into action. 5 seconds gives you enough time to evaluate what the TCAS is telling you to do; i.e.; climb; descend and how much; or stay where you are. At high altitudes; it is easy to over control; especially when all you want is a 300 FPM climb or descent. The TCAS also reads a climbing or descending trend; even a small one of 100 FPM such as found in mild mountain wave. With separation at altitude so small (1;000 FT) an RA can happen in a heart beat even though separation limits are not exceeded. Supplemental information from ACN 823379: We were questioned by ZMP as to why our aircraft responded to the RAs when the Controller had both airliners level at their assigned altitudes. I wish I could provide specific action that would prevent this from happening in the future; but I am not certain of the cause. Possible slight wave action combined with equipment sensitivity may have caused this event.