1.6 KiB
Case 1: if the critical point is hyperbolic, life is okay. Linearize about that point, look at eigenvalues and eigenvectors to understand our different manifolds. Case 2: If the point is NOT hyperbolic. We've got to do something else.
Assume \vec{P} \in R^3 is a critical point in our system \dot{X} = F(x), x\in R^3
Define stable and unstable manifolds of that point P as:
W_s(\vec{P}) = \left\{x: \Lambda^+(x) = \vec{P} \right\}
W_u(\vec{P}) = \left\{x: \Lambda^-(x) = \vec{P} \right\}
Where the first is forward in time, the second is backward in time.
Theorem: x is some some differential equation system in R^n and f = c^1(E) (c1 continuous over E, where E is an open subset of R^n, containing the origin)
If f(0)=0, the Jacobian has n eigenvalues with a nonzero real part. (Hyperbolic)! Then in a small neighborhood of x\approx 0 There exists stable and unstable manifolds of the linearized system \dot{x} = Jx
where J is the Jacobian, and W_s and W_u are tangent to E_s and E_u respectively at x=0. E defines the eigenspace.
What they hell do we do when eigenvalues do not have a real part?
Center Manifold: W_c and Center Eigenspace: E_c. Where the same rules apply as above. W_c is not generally unique.
Center Manifold Theorem:
Let f \in C^1(E), r\leq1 where E is an open subspace of R^n .
If f(0)=0 and J has n_s eigenvalues with negative real part, n_u eigenvalues with positive real part, and if n_c = n-n_s-n_u purely imaginary eigenvalues exist,
Then there exists an n_c dimensional center manifold W_c of a class C^r which is tangent to E_c.
Examples in class slides.