Invariant Manifolds

  According to our discussion of invariant sets in section 4.3.1, we would like to analyze a dynamical system by breaking it into its dynamically invariant parts. This is particularly easy to accomplish with linear systems because we can write down a general solution for the flow operator as (see section 4.5.2). The eigenspaces of a linear flow or map (i.e., the spaces formed by the eigenvectors of A) are invariant subspaces of the dynamical system. Moreover, the dynamics on each subspace are determined by the eigenvalues of that subspace. If the original manifold is , then each invariant subspace is also a Euclidean manifold which is a subset of . It is sensible to classify each of these invariant submanifolds according to the real parts of its eigenvalues, :

is called the stable space  of dimension , is called the center space  of dimension , and is called the unstable space  of dimension . If the original linear manifold is of dimension n, then the sum of the dimensions of the invariant subspaces must equal n: . This definition also works for maps when the conditions on are replaced by modulus less than one ( ), modulus equal to one ( ), and modulus greater than one ( ).

For example, consider the matrix

This matrix has eigenvalues and eigenvectors (1,-1,0), (0,0,1), (2,1,0), and the flow on the invariant manifolds is illustrated in Figure 4.10.

  
Figure 4.10: Invariant manifolds for a linear flow with eigenvalues .

• Center Manifold Theorem
• Homoclinic and Heteroclinic Points