  In this article, we focus on the invariance property of Morse homology on
noncompact manifolds. We expect to apply outcomes of this article to several
types of Floer homology, thus we define Morse homology purely axiomatically and
algebraically. The Morse homology on noncompact manifolds generally depends on
the choice of Morse functions; it is easy to see that critical points may
escape along homotopies of Morse functions on noncompact manifolds. Even worse,
homology classes also can escape along homotopies even though critical points
are alive. The aim of the article is two fold. First, we give an example which
breaks the invariance property by the escape of homology classes and find
appropriate growth conditions on homotopies which prevent such an escape. This
takes advantage of the bifurcation method. Another goal is to apply the first
results to the invariance problem of Rabinowitz Floer homology. The bifurcation
method for Rabinowitz Floer homology, however, is not worked out yet. Thus
believing that the bifurcation method is applicable to Rabinowitz Floer
homology, we study the invariance problems of Rabinowitz Floer homology.
