The Israelis dont care about Syrian or any other Arab coutry's stamp in your passport. They will search your luggage very seriously though.they will not stqmp your pqssport for the entry, but they will stamp it for the bike. no worries there though, it is a totally nondescript stamp with a number on it, that's all.
Make sure the Jordanians though when you leqve do NOT stamp you OUT. insteqd, they give you a white peace of paper which you redeem when you reenter Jordan. So for all practical purpose, it is as if you never left the country... I strongly suggest that when you arrive in JOrdan from Syria, you go straight to the main police station in Amman to get the three month extention on your jordan visa. it's free. That way, the Syrians, on the way out, will think you stayed in Jordan all that time.
Where things get more complicated is on the way back, the Jordanians will only issue you a transit permit for the bike. you can solve this problem in Amman by getting an extention on your bike permit. They will do it, but it is time consuming bureaucracy. if you forget, it is not a big prblem, they will fine you 5 jordanian dinars upon exit.
hope this helps