Center Manifold Theory
Center manifold theory is essential for analyzing local bifurcations. As the Liapunov-Schmidt reduction for stationary and Hopf bifurcations, center manifold theory is used to reduce a dynamical system near a nonhyperbolic equilibrium or a periodic solution to a low-dimensional system with the vector field as functions of the critical modes. Furthermore, stability of solutions and local dynamics of the system can be derived from the low-dimensional system. The center manifold theorem was introduced in the sixties by Pliss  and Kelley . Owing to the Lanford’s contribution  this theory has been applied extensively to the study of bifurcation problems and dynamical systems, in particular, in connection with the normal form theory.
KeywordsHopf Bifurcation Taylor Expansion Bifurcation Point Critical Mode Center Manifold
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