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Unformatted text preview: Center Manifold Theory CDS140B Lecturer: Wang Sang Koon Winter, 2004 Introduction to Bifurcation Thoery. In this chapter, we will cover the following materials: • Center Manifold Theory allows us to reduce the dimension of a problem, you will most likely still be left with a nonlinear system. • Normal Form Theory can be used to “simplify” the nonlinear system by (removing as much nonlinearity as possible. This involves nonlinear coordinate transformation. • Local Bifurcation Theory uses the above techniques to determine when the system changes qualitatively as parameters are varied. 1 Center Manifold Theory 1.1 Existence Theorem 13.3 (Existence). Consider ˙ x = Ax + f ( x ) where 1. x ∈ R n and A is a constant n × n matrix; x = 0 is an isolated critical point; the vector function f ( x ) is C k ,k ≥ 2, in a neighborhood of x = 0 and lim || x ||← || f ( x ) || / || x || = 0; 2. the stable and unstable manifolds of equation ˙ y = Ay are E s and E u , the space of eigenvectors corresponding with eigenvalues with zero real part...
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- Fall '10
- Manifold, Fundamental physics concepts, Center manifold, Bifurcation Thoery