Cylindropuntia arbuscula perhaps? http://www.desertmuseumdigitallibrary.org/public/detail.php?id=ASDM00866&sp=Cylindropuntia arbuscula http://www.fireflyforest.com/flowers/others/other23.html
Thank you so much, Saltcedar! I am very excited, because I went to the links you provided, and their photos exactly match what I am seeing in my yard! The flowers are identical to what I have! Thank you so much for your help identifying my cactus! This forum is awesome, and your contributions to my identification questions are very much appreciated! Thanks so much for your time!