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.
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