Please indicate a reference (if there exists any) proving that when a cycle appears (i.e. for the minimal value of the logistic parameter $r$ for which a $k$-cycle exists) we also have, "immediately", stability of this/these cycle(s). Whenever I read about this, it appears a "known" fact, or it is noticed as being "expected".
Also, is there a theoretical result which proves that if there exist several $k$-cycles for a given $r$ there is at most one that is stable?