The stable manifold theorem

We now consider systems which have stable as well as unstable eigenvalues. Consider the system

   $$\begin{eqnarray} \dot x&=&Ax+f(x,y),\nonumber\ \dot y&=&By+g(x,y),\end{eqnarray}$$

where $x\in{\rm I\!R}^n$, $y\in{\rm I\!R}^m$, all eigenvalues of A are in the left half plane, all eigenvalues of B are in the right half plane, and the nonlinearities satisfy

$$\Vert f(x,y)\Vert+\Vert g(x,y)\Vert\le C(\Vert x\Vert^2+\Vert y\Vert^2)$$ (77)

for small x and y.

For the linear system, there is an n-dimensional space (x arbitrary and y=0) in which solutions approach zero as $t\to\infty$, and an m-dimensional space (x=0 and y arbitrary) in which solutions approach zero as $t\to -\infty$). An example of this was shown in Figure 1.3, where there is a line along which solutions move towards the origin, and another line along which they move away from the origin. The stable manifold theorem says that the nonlinear system behaves in a qualitatively similar fashion; the only difference is that the linear subspaces must be replaced by curved manifolds.

Theorem 5

In some neighborhood of the origin, there exist unique and smooth functions $y=\phi(x)$ and $x=\psi(y)$ with the following properties:
1. The manifolds $y=\phi(x)$ and $x=\psi(y)$ are invariant, i.e. if a solution of (2.24) starts on one of these manifolds, then it remains there.
2. The manifolds are tangent to the spaces y=0 and x=0, respectively, i.e. $\Vert\phi(x)\Vert=O(\Vert x\Vert^2)$ and $\Vert\psi(y)\Vert=O(\Vert y\Vert^2)$ as $x,y\to 0$.
3. A solution of (2.24) approaches the origin as $t\to\infty$ precisely if it lies on the manifold $y=\phi(x)$, and it approaches the origin as $t\to -\infty$ precisely if it lies on the manifold $x=\psi(y)$.

The manifold $y=\phi(x)$ is called the stable manifold and the manifold $x=\psi(y)$ is called the unstable manifold.

The existence of an unstable manifold is one avenue towards proving that linear instability implies nonlinear instability. This appears intuitively obvious, but is actually harder to prove than one might expect.

Figure 2.1 shows a plot of the direction field for the system $\dot x_1=x_2+x_1^2$, $\dot x_2=x_1+x_2^2+x_1x_2$. The stable manifold is tangent to the line x₁=-x₂, and the unstable manifold is tangent to the line x₁=x₂ at the origin. On one side of the origin, the unstable manifold goes to another equilibrium point, which is roughly at (-0.755,-0.570).

 

Figure 2.1: Phase vector diagram for $\dot x_1=x_2+x_1^2$, $\dot x_2=x_1+x_2^2+x_1x_2$. 

There is also a version of the stable manifold theorem which covers the case where some of the eigenvalues are on the imaginary axis. Consider the system

   $$\begin{eqnarray} \dot x&=&Ax+f(x,y,z),\nonumber\ \dot y&=&By+g(x,y,z),\nonumber\ \dot z&=&Cz+h(x,y,z),\end{eqnarray}$$

where f, g and h are of quadratic order at the origin, A has eigenvalues with negative real parts, B has eigenvalues with positive real parts, and C has eigenvalues with zero real part. The stable manifold in this case has the form $y=\phi_1(x)$, $z=\phi_2(x)$, and the unstable manifold has the form $x=\psi_1(y)$, $z=\psi_2(y)$, where, as before, the $\phi_i$ and $\psi_i$are of quadratic order at the origin. The difference is that we can no longer characterize the stable manifold as the locus of solutions which approach the origin for $t\to\infty$. This is because of the presence of neutral eigenvalues. For instance, the third equation of (2.26) might read $\dot z=-z^2$, and in this case every solution with positive initial condition will approach zero as $t\to\infty$, but solutions grow if z is negative. In the presence of neutral eigenvalues, therefore, the nonlinear terms determine which solutions approach the origin, and these solutions need not even form a manifold. The proper characterization of the stable manifold for this case is the locus of all solutions which approach the origin exponentially as $t\to\infty$.