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Unformatted text preview: Bifurcation of Fixed Points CDS140B Lecturer: Wang Sang Koon Winter, 2005 1 Introduction Consider y = g ( y, ) . where y R n , R p . Suppose it has a fixed point at ( y , ), i.e., g ( y , ) = 0. Two Questions: (1) Is the fixed point stable or unstable? (2) How is the stability or instability affected as is varied? Hyperbolic Fixed Points. None of the eigenvalues of D y g ( y , ) lie on the the imaginary axis. The stability of ( y , ) is determined by its linearized equation. Since hyperbolic fixed points are structurally stable, varying slightly does not change the nature of the stability of the fixed point. Nonhyperbolic Fixed Points. D y g ( y , ) has some eigenvalues on the imaginary axis. For very close to (and for y very close to y ), radically new dynamical behavior can occur: fixed points can be created or destroyed, and periodic, quasiperiodic, or even chaotic dynamics can be created. We will begin by studying the case where the linearized equation has a single zero eigenvalue with the remaining eigenvalues having nonzero real parts. 1.1 A Zero Eigenvalue In this case, the orbit structure near ( y , ) is determined by the associated center manifold equation x = f ( x, ) where x R 1...
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
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