In the previous subsection, we used the words ‘stable’ and ‘unstable’ without giving a careful definition of what those terms mean. We know that for homogeneous constant-coefficient systems x0 = Ax, the stability of the equilibrium at the origin is somehow determined by the eigenvalues of A. The eigenvectors of A determine how the phase portrait is “skewed” from one of the three canonical phase portraits.
We will give a rigorous definition of stability of equilibria in the next chapter;
for now, a loose definition will suffice. An equilibrium solution x∗ of a system of ODEs is stable if, whenever we start from initial conditions that are appropriately
“close” to x∗, the resulting solution trajectory never strays too far from x∗. A stable equilibrium is called asymptotically stable if the solution trajectory actually approaches x∗ as t → ∞. If an equilibrium is not stable, it is called unstable.
This means that there exist initial conditions arbitrarily “close” to x∗ for which the solution trajectory is repelled from x∗.
For example, if the origin is a center, then we would say the origin is a stable equilibrium but is not asymptotically stable. Saddle equilibria are unstable, because it is always possible to choose initial conditions arbitrarily close to the equilibrium for which the direction of motion in the phase portrait is away from the equilibrium as t increases. (Just choose any initial conditions not lying on the separatrix that is oriented towards the equilibrium.)
Saddles are interesting in that there are “special” trajectories in the phase portrait on which the direction of motion is directed toward the unstable equilibrium. In our canonical example of a saddle, one separatrix was oriented toward the origin, and the other separatrix was oriented away from the origin. The notion that our underlying space can be decomposed into stable and unstable “directions” is the subject of our discussion below. First, we recall a familiar definition from linear algebra.
Definition 2.3.1. Let v1, v2, . . . , vk be vectors in Rn. The span of these vectors is the set of all linear combinations
c1v1+ c2v2+ · · · + ckvk, where c1, c2, . . . ck are real numbers.
Notice that if these vectors are linearly independent, then their span forms a k-dimensional subspace of Rn.
Example. The span of the vectors
1 0 0
and
1 1 0
in R3 is the xy-plane.
Now consider the homogeneous system x0 = Ax of ODEs, where A is an n × n matrix. Let λ1, λ2, . . . λndenote the eigenvalues of A, repeated according to algebraic multiplicity. Each eigenvalue can be written in the form λj = αj+ iβj, where αj and βj are real. (Of course, βj = 0 if the eigenvalue λj is real.) Associated with each eigenvalue is a set of eigenvectors (and possibly generalized eigenvectors).
Definition 2.3.2. The stable subspace of the system x0 = Ax is the span of all eigenvectors and generalized eigenvectors associated with eigenvalues having negative real part (αj < 0). The unstable subspace of the system x0 = Ax is the span of all eigenvectors and generalized eigenvectors associated with eigenvalues having positive
eigenvectors and generalized eigenvectors associated with eigenvalues having zero real part (αj = 0).
Notation: The stable, unstable, and center subspaces are denoted by Es, Eu, and Ec, respectively.
Letting A denote the coefficient matrix, the characteristic equation is given by λ2− 5λ − 6 = 0.
Factoring the characteristic equation as (λ − 6)(λ + 1) = 0, we obtain eigenvalues λ = −1 and λ = 6. The roots are real and have opposite sign, indicating that the origin is a saddle. You can verify that
· −1
are eigenvectors corresponding to λ = −1 and λ = 6, respectively. By the above definition, the stable subspace is given by
Es = span
½· −1 1
¸¾
and the unstable subspace is given by Eu = span
½· 2 15
¸¾
The center subspace Ec consists only of the zero vector. Notice that Es and Eu really are subspaces of R2, the underlying space. They are straight lines through the origin and, in this example of a saddle, they correspond to the separatrices (see Figure 2.11).
We remark that the general solution of this system is given by
· x1(t)
If we start from (non-zero) initial conditions inside the stable subspace (c1 6= 0 and c2 = 0), then our solution trajectory will remain in the stable subspace for all time
x2
x1
Es Eu
Figure 2.11: Phase portrait showing the stable and unstable subspaces Es and Eu. t, and we will approach the origin as t → ∞. Likewise, if we start from initial conditions inside the unstable subspace, (c1 = 0 and c2 6= 0), then our solution trajectory remains in the unstable subspace for all time t but we always move away from the origin.
Example. The system
x0 =
· 1 −8
8 1
¸ x
has a coefficient matrix that is already in real canonical form, and its eigenvalues are 1 ± 8i. Since both eigenvalues have positive real part, Eu = R2 whereas both Es and Ec consist only of the zero vector. Likewise, the system
x0 =
· 0 −8
8 0
¸ x
has a coefficient matrix with eigenvalues ±8i, both of which have zero real part. In this case, Ec = R2 while the stable and unstable subspaces consist only of the zero vector.
The notions of stable, unstable, and center subspaces are not restricted to planar systems, as we illustrate in the following example.
Example. Consider the system
x0 =
−2 −1 0 1 −2 0
0 0 8
x,
whose coefficient matrix has a convenient block-diagonal structure. The eigenvalues are −2 ± i and 8. The eigenvalue −2 + i gives rise to a complex eigenvector
and the real eigenvalue has
0 0 1
as an eigenvector. Since −2 + i has negative real part, we conclude that the stable subspace is Graphically, this is the xy-plane in R3. The unstable subspace
Eu = span
is one-dimensional and corresponds to the z-axis in R3. The center subspace Ec consists only of the zero vector.
Several observations will help us sketch the phase portrait. Any trajectory start-ing from non-zero initial conditions in Es (the xy-plane) will spiral inward toward the origin while always remaining within Es. Any non-zero trajectory starting in Eu (the z-axis) will remain within Eu for all time t and will be oriented outward from the origin. Finally, any trajectory starting outside Es and Eu will spiral away from the xy-plane but will draw closer to the z-axis as t advances. A sketch of the phase portrait appears in Figure 2.12.
As another illustration, the system
x0 =
has a coefficient matrix with eigenvalues ±i and 8. The corresponding eigenvectors are exactly as in the previous example. The only difference is that the xy-plane is now the center subspace because the eigenvectors that span that plane correspond to an eigenvalue with zero real part. The z-axis is the unstable subspace. Any trajectory starting from non-zero initial conditions in Ec (the xy-plane) will circle around the
Es Eu
Figure 2.12: Phase portrait of a 3-D system with a two-dimensional stable subspace Es and a one-dimensional unstable subspace Eu.
Ec Eu
Figure 2.13: Phase portrait of a 3-D system with a two-dimensional center subspace Ec and one-dimensional unstable subspace Eu.
origin, remaining in Ecfor all time t without being attracted or repelled by the origin.
Any non-zero trajectory starting in Eu(the z-axis) will remain within Eu for all time t and will be oriented outward from the origin. Finally, any trajectory starting outside Ec and Eu will spiral away from the xy-plane but will always maintain a constant distance from the z-axis as t advances. In other words, such trajectories are confined to “infinite cylinders.” A sketch of this phase portrait appears in Figure 2.13.
Before moving on, we introduce some important terminology related to our above discussion. As usual, consider the system x0 = Ax where A is an n × n constant matrix. Given an initial condition x0 = x(0), we know that the unique solution of this initial value problem is given by x(t) = etAx0. In terms of the phase portrait, once we pick the point x0, pre-multiplying by the matrix etA and letting t increase will trace out a curve in Rn. Let φt be the function which associates each different
Definition 2.3.3. The set of functions φt = etA is called the flow of the system x0 = Ax of ODEs.
The reason for using the name “flow” is that φt describes the motion along tra-jectories in the phase space starting from various choices of initial conditions x0. Definition 2.3.4. If all eigenvalues of A have non-zero real part, the flow is called a hyperbolic flow the system x0 = Ax is called a hyperbolic system, and the origin is called a hyperbolic equilibrium point.
Example. The system x0 = Ax where A =
· 0 1
−9 0
¸ ,
is non-hyperbolic because the eigenvalues of A are λ = ±3i, both of which have zero real part. The origin, a center in this case, is a non-hyperbolic equilibrium. Note that, if an equilibrium is hyperbolic, then the dimension of its center subspace is zero.
Our next comments concern properties of the stable, unstable, and center sub-spaces associated with the system x0 = Ax. In the examples we gave, there are several common themes. First, the pairwise intersections of Es, Eu, and Ec consist only of the zero vector. Second, the sum of the dimensions of these three subspaces is always equal to the dimension of the underlying space Rn. Finally, if we start from an initial condition x0 which lies inside one of these three subspaces, then φt(x0) remains within that subspace for all real t. These observations are now stated formally.
Recall that if S1 and S2 are subspaces of a vector space V , then the sum of the subspaces is defined as
S1+ S2 = {x + y : x ∈ S1 and y ∈ S2} .
The sum S1+ S2 is itself a subspace of V , and the concept of sums of vector spaces is easily extended to larger finite sums. In the special case where S1 + S2 = V and S1∩ S2 consists only of the zero vector, we refer to the sum as a direct sum and write S1⊕ S2 = V .
Theorem 2.3.5. Consider the system x0 = Ax, where A is an n×n constant matrix, and let Es, Eu, and Ec denote the stable, unstable, and center subspaces associated with the equilibrium at the origin. Then Rn = Es⊕ Eu⊕ Ec.
Theorem 2.3.6. The subspaces Es, Eu and Ec are invariant with respect to the flow φt = etA in the following sense: If x0 is any initial condition in Es, then etAx0
is in Es for all t. Similar statements hold for Ec and Eu.
In the phase portrait, if we start from within one of these three invariant subspaces, our solution trajectory will never escape from that subspace.
The final theorem in this section is one of the most important qualitative results for the constant-coefficient systems we have studied up to now. It tells us what sort of behavior we can expect if we know the eigenvalues of the coefficient matrix A.
Theorem 2.3.7. Consider the linear system x0 = Ax, where A is an n × n constant matrix. Then the following statements are equivalent:
• Given any x0 ∈ Rn, we have
t→∞lim etAx0 = 0,
and for any non-zero x0 ∈ Rn, the distance from etAx0 to the origin tends to ∞ as t → −∞.
• All eigenvalues of A have negative real part.
• The stable subspace Es at the origin is the entire space Rn.
In other words, if all eigenvalues have negative real part, then the flow directs us towards the origin as t → ∞ and we are repelled from the origin as t → −∞. The proofs of the three preceding theorems are not difficult, but are omitted.
For linear, homogeneous, constant-coefficient systems, it is easy to classify the stability of the origin. Namely, the origin is
• Asymptotically stable if all eigenvalues of A have negative real part.
• Stable if none of the eigenvalues has positive real part.
• Unstable if any of the eigenvalues has positive real part.
Example. If
A =
· 1 0 0 −1
¸ ,
then the origin is unstable because one of the eigenvalues is positive. If A =
· 0 1
−1 0
¸ , then the origin is stable but not asymptotically stable. If
A =
· −1 0 0 −6
¸ , then the origin is asymptotically stable.