A survey on foundation of bifurcation theory
Date Issued
2010
Date
2010
Author(s)
Liang, Chun-Ming
Abstract
Many problems in mathematics, and its applications to theoretical physics, lead to a problem of the form
f(λ,x)=0, (1)
where f is an operator on R×X into Y , and X and Y are Banach spaces. For example, (1) could represent a system of differential or integral equations, depending on a parameter λ. We are interested in the structure of the solution set; namely, the set
f^(-1) (0)={ (λ,x)∈R×X∶f(λ,x)=0 }. (2)
Since we are interested in bifurcation from trivial solutions, we may assume that (λ,0) is a solution curve of (1). In particular, we seek conditions on f to see if a solution (λ,0) of (1) whether or not lies on the other solution curves of (1). In this paper, we start with introducing the bifurcation theory in finite dimensional space case. The degree theory is used in both finite and the infinite dimensional space cases. We conclude the article with some examples.
Subjects
bifurcation
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