2.7 the Stable Manifold Theorem

The stable manifold theorem is one of the most important results in the local qualitative theory of ordinary diﬀerential equations. The theorem shows that near a hyperbolic equilibrium point x0, the nonlinear system x˙ = f(x) (1) s has stable and unstable manifolds S and U tangent at x0 to the stable and unstable subspaces E and Eu of the linearized system x˙ = Ax, (2) s u where A = Df(x0). Furthermore, S and U are of the same dimensions as E and E , and if φt is the ﬂow of the nonlinear system (1), then S and U are positively and negatively invariant under φt respectively and satisfy lim φt(c) = x0 for all c ∈ S t→∞ and lim φt(c) = x0 for all c ∈ U. t→−∞ Example 1. Consider the system

2 2 x˙1 = −x1, x˙2 = −x2 + x1, x˙3 = x3 + x1. The only equilibrium point of this system is at the origin (0, 0, 0).

−1 0 0 Df = 2x1 −1 0 . 2x1 0 1

−1 0 0 Thus A = Df(0, 0, 0) =  0 −1 0 has eigenvalues λ1 = −1 = λ2 and λ3 = 1 and hence (0, 0, 0) 0 0 1 1 is a hyperbolic equilibrium point. The eigenvectors corresponding to λ1 = −1 = λ2 are u1 = 0 0 0 and u2 = 1. 0 0 The eigenvector corresponding to λ3 = 1 is u3 = 0. Thus, the stable and unstable subspaces 1 s u E and E of the system are the x1, x2 plane and the x3−axis respectively. The solution of the system is given by

−t x1(t) = c1e −t 2 −t −2t x2 = c2e + c1(e − e ) t 1 2 t −2t x3(t) = c3e + 3 c1(e − e ), where x(0) = c. 1 2 Clearly, lim φt(c) = 0 if and only if c3 + c1 = 0. Thus, the stable manifold is t→∞ 3 1 S = {c ∈ 3 : c + c2 = 0}. R 3 3 1 Similarly, lim φt(c) = 0 if and only if c1 = c2 = 0. Thus, the unstable manifold is t→−∞

3 U = {c ∈ R : c1 = c2 = 0}. The stable and unstable manifolds for this system are shown in Figure 1. Note that the surface s u S is tangent to E , i.e., to the x1, x2 plane at the origin and that U = E .

Deﬁnition. (Homeomorphysim ) Let X be a metric space and let A and B be subsets of X. A homeomorphism of A onto B is a continuous one-to-one map of A onto B, h : A → B, such that h−1 : B → A is continuous. The sets A and B are called homeomorphic or topologically equivalent if there is a homeomorphism of A onto B

Definition. (Differentiable manifold ) An n−dimensional differentiable manifold, M (or a manifold of class Ck), is a connected metric space S with an open covering {Uα}, i.e., M = α Uα, such that

n n 1. for all α, Uα is homeomorphic to the open unit ball in R , B = {x ∈ R : |x| < 1}, i.e., for all α there exists a homeomorphism of Uα onto B, hα : Uα → B, and T T 2. if Uα Uβ 6= ∅ and hα : Uα → B, hβ : Uβ → B are homeomorphisms, then hα(Uα Uβ) and T n hβ(Uα Uβ) are subsets of R and the map

−1 \ \ h = hα ◦ hβ : hβ(Uα Uβ) → hα(Uα Uβ)

k T is diﬀerentiable (or of class C ) and for all x ∈ hβ(Uα Uβ), the Jacobian determinant detDh(x) 6= 0.

−1 The manifold M is said to be analytic if the maps h = hα ◦ hβ are analytic.

2 The cylindrical surface S in the above example is a two-dimensional differentiable manifold. The projection of the x1x2 plane onto S maps the unit disks centered at the points (m, n) in the x1x2 2 2 2 plane onto homeomorphic images of the unit disk B = {x ∈ R : x1 + x2 < 1}. These sets Umn ⊆ S then form a countable open cover of S in this case. The pair (Uα, hα) is called a chart for the manifold M and the set of all charts is called an atlas for −1 M. The differentiable manifold M is called orientable if there is an atlas with det Dhα ◦ h (x) > 0 T β for all α, β and x ∈: hβ(Uα Uβ). Theorem. (The Stable Manifold Theorem) n 1 Let E be an open subset of R containing the origin, let f ∈ C (E), and let φt be the flow of the nonlinear system (1). Suppose that f(0) = 0 and that Df(0) has k eigenvalues with negative real part and n − k eigenvalues with positive real part. Then there exists a k−dimensional differentiable manifold S tangent to the stable subspace Es of the linear system (2) at 0 such that for all t > 0, φt(S) ⊆ S and for all x0 ∈ S, lim φt(x0) = 0; t→∞ and there exists an n − k dimensional differentiable manifold U tangent to the unstable subspace Eu of (2) at 0 such that for all t < 0, φt(U) ⊆ U and for all x0 ∈ U,

lim φt(x0) = 0. t→−∞

Finding the stable and unstable manifolds

We remark that if f ∈ C1(E), and f(0) = 0, then the system (1) can be written as

x˙ = Ax + F (x), (3) where A = Df(0),F (x) = f(x) − Ax, F ∈ C1(E),F (0) = 0 and DF (0) = 0. Furthermore, as in Section 1.8 of Chapter 1, there is an n × n invertible matrix C such that

P 0 B = C−1AC = , 0 Q where the eigenvalues λ1, ··· , λk of the k × k matrix P have negative real part and the eigenvalues −1 λk+1, ··· , λn of the (n − k) × (n − k) matrix Q have positive real part. Letting y = C x, the system (3) then has the form y˙ = By + G(y), (4) where G(y) = C−1F (Cy) ∈ C1(E˜) where E˜ = C−1(E). It will can be shown that there are n − k differentiable functions ψj(y1, ..., yk) such that the equations yj = ψj(y1, ..., yk), j = k + 1, ..., n define a k−dimensional differentiable manifold S˜ in y−space. The differentiable manifold S in x−space is then obtained from S˜ under the linear transformation of coordinates x = Cy. Let eP t 0 0 0 U(t) = and V (t) . 0 0 0 eQt Then U˙ = BU, V˙ = BV and eBt = U(t) + V (t).

3 Consider the integral equation Z t Z ∞ u(t, a) = U(t)a + U(t − s)G(u(s, a))ds − V (t − s)G(u(s, a))ds. 0 t If u(t, a) is a continuous solution of this integral equation, then it is a solution of the diﬀerential equation (4). We now solve this integral equation by the method of successive approximations. Let

u(0)(t, a) = 0 and Z t Z ∞ u(j+1)(t, a) = U(t)a + U(t − s)G(u(j)(s, a))ds − V (t − s)G(u(j)(a, a))ds. 0 t For j = k + 1, ..., n we deﬁne the functions

ψj(a1..., ak) = uj(0, a1, ..., ak, 0, ..., 0).

Then the initial values yj = uj(0, a1, ..., ak, 0, ..., 0) satisfy

yj = ψj(y1, ··· , yk) for j = k + 1, ..., n. These equations then deﬁne a diﬀerentiable manifold S˜. The existence of the unstable manifold U of (4) is established in exactly the same way by con- sidering the system (4) with t → −t, i.e.,

y˙ = −By − G(y).

The stable manifold for this system will then be the unstable manifold U for (4). Note that it is also necessary to replace the vector y by the vector (yk+1, ··· , yn, y1, ··· , yk) in order to determine the n − k dimensional manifold U by the above process. Example 1. Consider 2 x˙1 = x1 + 4x2 − x1, x˙2 = 6x1 − x2 + 2x1x2. 1 4 The critical point is (0, 0) and A = Df(0, 0) = . The eigenvalues of A are λ = −5, λ = 5 6 −1 1 2 2 1 and the corresponding eigenvectors are u = and u = . Thus 1 −3 2 1

2 1 C = . −3 1 Example 2. Consider the nonlinear system

2 2 x˙1 = −x1 − x2, x˙2 = x2 + x1. We shall ﬁnd the ﬁrst three successive approximations u(1)(t, a), u(2)(t, a) and u(3)(t, a) and use (3) u (t, a) to approximate the function ψ2 describing the stable manifold

S : x2 = ψ2(x1). For this problem, we have

2 −1 0 −x2 A = B = ,F (x) = G(x) = 2 , 0 1 x1

4 e−t 0 0 0 U(t) = ,V (t) = 0 0 0 et a and a = 1 . We approximate the solution of the integral equation 0

−t Z t Z ∞ a1e s−t 2 0 u(t, a) = + −e u2(s) ds − t−s 2 ds. 0 0 t e u1(s)

a e−t by the successive approximations u(0)(t, a) = 0, u(1)(t, a) = 1 , 0

−t (2) a1e u (t, a) = 1 2 −2t , − 3 a1e

−t 1 4 −4t −t (3) a1e + 27 a1(e − e ) u (t, a) = 1 2 −2t . − 3 a1e ∼ (3) If we approximate u(t, a) by u(t, a) = u (t, a), then ψ1(a1) = u1(0, a1, 0), ψ2(a1) = u2(0, a1, 0) are approximated by 1 ψ (a ) = a ψ (a ) = − a2 1 1 1 2 1 3 1 ˜ as a1 → 0. Hence, the stable manifold S is approximated by 1 S˜ : y = − y2 2 3 1 as y1 → 0. The matrix C = I, for this example and hence the x and y spaces are the same; that is the stable manifold S is approximated by 1 S : x = − x2 2 3 1 . The unstable manifold U can be approximated by applying exactly the same procedure to the above system with t → −t and xl and x2 interchanged. The stable manifold for the resulting system will then be the unstable manifold for the original system. We find 1 U : x = − x2 1 3 1 as x2 → 0. The stable and unstable manifolds S and U are only defined in a small neighborhood of the origin in the proof of the stable manifold theorem. S and U are therefore referred to as the local stable and unstable manifolds of (1) at the origin or simply as the local stable and unstable manifolds of the origin. We define the global stable and unstable manifolds of (1) at 0 by letting points in S flow backward in time and those in U flow forward in time.

Definition. ( Global stable and unstable manifolds) Let φt be the flow of the nonlinear system (1). The global stable and unstable manifolds of (1) at 0 are defined by s [ W (0) = φt(S) t≤0

5 and u [ W (0) = φt(U) t≥0 respectively; W s(0) and W u(0) are also referred to as the global stable and unstable manifolds of the origin respectively.

Theorem. (The Center Manifold Theorem) Let E be an open subset of Rn containing the origin, let f ∈ Cr(E). Suppose that f(0) = 0 and that Df(0) has k eigenvalues with negative real part, j eigenvalues with positive real part and m = n−k−j eigenvalues with zero real part. Then there exists a m−dimensional manifold W c(0) of class Cr tangent to the center subspace Ec of (2) at 0, there exists a k−dimensional manifold W s(0) of class Cr tangent to the stable subspace Es of the linear system (2) and there exists an j−dimensional diﬀeren- tiable manifold W u(0) tangent to the unstable subspace Eu of (2) at 0. Furthermore, W s(0),W u(0) c and W (0) are invariant under the ﬂow φt of (1). Example 1. Consider 2 x˙1 = x1, x˙2 = −x2. 0 0 The critical point is (0, 0) and A = Df(0, 0) = . The eigenvalues of A are λ = 0, λ = −1 0 −1 1 2 1 0 and the corresponding eigenvectors are u = and u = . The stable subspace Es of the 1 0 2 1 c linearized system at the origin is the x2−axis and the center subspace E is the x1−axis. This system has the solution c1 −t x1(t) = , x2(t) = c2e . 1 − c1t The phase portrait is shown in Figure 5

6 Any solution curve of the system to the left of the origin patched together with the positive ∞ c x1−axis at the origin gives a one dimensional center manifold of class C which is tangent to E at the origin. This shows that, in general, the center manifold W c(0) is not unique; however, in this example there is only one analytic center manifold, namely, the x1−axis.

Problems: 1—5