ABSTRACT
In Chapter 4, Theorem 4.11, we were able to give a complete determination of the stability of two-dimensional maps via linearization, when the fixed point is hyperbolic, i.e., the eigenvalues of the Jacobian matrix are off the unit circle. However, Theorem 4.11 failed to address the stability of nonhyperbolic fixed points. Here comes the center manifold theory to our rescue. Roughly speaking, a center manifold is a setMc in a lower dimensional space, where the dynamics of the original system can be obtained by studying the dynamics on Mc. For instance, if a nonhyperbolic map is defined on R2, then its dynamics may be analyzed by studying the dynamics on an associated one-dimensional center manifold Mc. In light of our complete understanding of the stability of one-dimensional maps, the reduction procedure to center manifolds is one of the most powerful tools in dynamical systems. We begin our exploration by introducing the necessary notations, definitions, theory and examples. Consider the s-parameter map F (µ, u), F : Rs × Rk → Rk, with u ∈ Rk,
µ ∈ Rs, where F is Cr (r ≥ 3) on some sufficiently large open set in Rk ×Rs. Let (µ∗, u∗) be a fixed point of F , i.e.,
F (µ∗, u∗) = u∗. (5.1)
We have seen in Chapters 1 and 4 that the stability of hyperbolic fixed points of F is determined from the stability of the fixed points under the linear map
However, the situation is drastically different if one of the eigenvalues λ of J lies on the unit circle, that is, |λ| = 1. There are three separate cases in which the fixed point (u0, µ0) is nonhyperbolic.
