A manifold is a generalization of the flat objects (lines, planes, and hyperplanes) that you learned about in linear algebra. The precise definition of a manifold is some-what technical and is not required for our purposes; instead we give some examples.
A smooth curve such as a circle in the plane is an example of a one-dimensional manifold. If we “zoom in” on any point within the circle, what we see would be indistinguishable from a small interval on the (one-dimensional) real line. Likewise, the surfaces in R3 you learned about in multi-variable calculus would be examples of two-dimensional manifolds. If we suitably magnify any tiny region on the graph of the paraboloid f (x, y) = x2+ y2, then the result would be virtually indistinguish-able from a region within the “flat” space R2. We shall deal only with differentiable manifolds (imagine a surface which does not have any sharp corners or ridges).
Below, we shall find that near hyperbolic equilibria, we expect nonlinear systems to have stable and unstable manifolds which have the same dimensions and invariance properties of Es and Eu for the linearized systems. First, we review the concept of the flow.
Suppose that x0 = f (x) and assume that all partial derivatives in Jf (x) are continuous for all x ∈ Rn. Given an initial condition x(0) = x0, Theorem 3.2.1 tells us that the initial value problem has a unique solution which exists in some open interval containing t = 0. Let φt(x0) denote the solution of this initial value problem, defined on the largest time interval containing t = 0 over which the solution exists.
Definition 3.6.1. The set of all such functions φt(x0) (i.e., for all possible choices of x0) is called the flow of the ODE x0 = f (x).
Notice that for each choice of x0, the function φt(x0) defines a parametrized curve
we discussed in the previous chapter, we know that the phase portraits may contain saddles, foci, nodes, and centers. The flows for nonlinear systems may exhibit much more interesting behavior.
Definition 3.6.2. A subset E ⊂ Rnis called invariant with respect to the flow if for each x0 ∈ E, the curve φt(x0) remains inside E for all time t over which the solution actually exists.
For linear, homogeneous, constant-coefficient systems x0 = Ax, the subspaces Es Ec, and Eu associated with the equilibrium at the origin are all invariant with respect to the flow. If we start from an initial condition inside any one of these subspaces, our solution trajectory remains confined to that subspace for all time t.
We now state another major theorem which, together with the Hartman-Grobman Theorem, provides much of our basis for understanding the qualitative behavior of solutions of nonlinear ODEs.
Theorem 3.6.3. (Stable Manifold Theorem). Suppose that x0 = f (x) is a sys-tem of ODEs for which the Jacobian matrix Jf (x) consists of continuous functions.
Further suppose that this system has an isolated, hyperbolic equilibrium point at the origin, and that the Jacobian Jf (0) has k eigenvalues with negative real part and (n − k) eigenvalues with positive real part. Then
• There is a k-dimensional differentiable manifold Ws which is (i) tangent to the stable subspace Es of the linearized system x0 = Jf (0)x at the origin; (ii) is invariant with respect to the flow; and (iii) for all initial conditions x0 ∈ Ws, we have
t→∞lim φt(x0) = 0.
• There is an (n−k)-dimensional differentiable manifold Wu which is (i) tangent to the unstable subspace Eu of the linearized system x0 = Jf (0)x at the origin;
(ii) is invariant with respect to the flow; and (iii) for all initial conditions x0 ∈ Wu, we have
t→−∞lim φt(x0) = 0.
Here, Ws and Wu are called the stable and unstable manifolds, respectively.
Remarks: Although the statement of this Theorem is a bit wordy, it is not difficult to understand what is going on. Basically, it says that near a hyperbolic, isolated equilibrium point, nonlinear systems produce objects Ws and Wu which are “curvy versions” of the subspaces Es and Eu for the linearized system. The manifold Ws is tangent to the subspace Es at the equilibrium. In the previous example, Ws was a parabola which was tangent to Es (the x-axis) at the origin (see Figure 3.2). If we start from an initial condition inside Ws, we will stay inside Ws for all time t, and we will approach the equilibrium as t → ∞. Similar statements hold for the unstable manifold.
Example. In the example we did in the previous section x0 = −x y0 = x2+ y, we noted that this nonlinear system has an exact solution
x(t) = x0e−t, y(t) = µ
y0+1 3x20
¶ et−1
3x20e−2t.
The origin is the only equilibrium solution, and we already established that it is a saddle. To determine Ws, we need to determine which special choices of initial conditions (x0, y0) would cause us to approach the origin as t → ∞. In the equation for x(t), we find that
t→∞lim x0e−t = 0
regardless of our choice for x0. This imposes no restrictions on our choice of initial conditions. Taking the same limit in the equation for y(t) is much more interesting:
in order to guarantee that
t→∞lim
·µ y0+1
3x20
¶ et−1
3x20e−2t
¸
= 0 we must insist that
y0+1
3x20 = 0,
because the exponential function et will increase without bound as t → ∞. No other restrictions are necessary, because e−2t → 0 as t → ∞. Therefore, we conclude that the one-dimensional stable manifold is given by the parabola y = −x2/3. A similar argument shows that Wu is the y-axis. Notice that Ws is tangent to the stable subspace Es (the x-axis) at the equilibrium point.
Example. Consider the nonlinear system
x0 = −x + 3y2 y0 = −y z0 = 3y2+ z.
Solve this system and compute the stable and unstable manifolds for the equilibrium at the origin.
Solution: This is very similar to the previous example. By computing the lineariza-tion, you should convince yourself that the origin really is a hyperbolic equilibrium.
We see immediately that y(t) = y0e−t. Substituting this expression into the first equation, we obtain a linear, inhomogeneous equation
0 2 −2t
The variation of parameters technique applies. Using eR1 dt = et as an integrating factor, we find that
et(x0+ x) = 3y20e−t, or equivalently,
d dt
¡etx¢
= 3y20e−t. Integrating both sides,
etx = etx¯
¯t=0 + Z t
0
3y20e−sds = x0+ 3y02 ¡
−e−s¢¯¯t
0 = x0+ 3y02(1 − e−t).
Multiplying both sides by e−t,
x(t) = x0e−t + 3y20¡
e−t− e−2t¢ .
The equation for z is solved in a similar fashion, and the general solution of the system is
x = ¡
x0 + 3y02¢
e−t − 3y02e−2t y = y0e−t
z = ¡
z0+ y02¢
et − y20e−2t.
The stable manifold Ws consists of all initial conditions (x0, y0, z0) such that the flow guides us to the origin as t → ∞. That is,
t→∞lim φt(x0, y0, z0) = (0, 0, 0).
Notice that all of the exponential functions in the solution are decaying except for the et term in the z(t) equation. In order to guarantee that we approach the origin as t → ∞, we need to coefficient of et to be 0. This forces
z0+ y02 = 0.
If we graph z = −y2 in R3, the result is a two-dimensional manifold: a parabolic sheet (see Figure 3.3). Similarly, solutions in the unstable manifold must approach the origin as t → −∞. In this limit, we have et → 0 but e−t → ∞ and e−2t → ∞.
Requiring the coefficients of both e−tand e−2tto be 0, it must be the case that x0 = 0 and y0 = 0, while z0 remains free. Therefore, the unstable manifold Wu consists of all points on the z-axis.
Ws Wu
x
z
y
Figure 3.3: Sketch of the two-dimensional stable manifold z = −y2 and the one-di-mensional unstable manifold (z-axis) from the example in the text.
We remark that when calculating Ws and Wu, it was important to express the solution (x, y, z) of the ODEs in terms of the initial conditions (x0, y0, z0) as opposed to introducing purely arbitrary constants C1, C2, and C3.
Calculating Ws and Wu by hand is usually impossible. There are methods for obtaining successive approximations of these manifolds, but such techniques can be very tedious.