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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