9.7, the last semi-short summary (for a while)

Section 9.7 is entitled “Qualitative Analysis of Linear Systems.”

Here’s a quick recap of how to solve a linear system:

1. Find an integer p such that the nullspace of [A – λI]^p has dimension q.

2. Find the basis of the nullspace of [A – λI]^p in the form {v₁, v₂,…, v_n}

3. For each v_j (where 1 ≤ j ≤ q), we have an equation of the form

Something we can notice about our solution procedure from the previous section is that we’ll have a linear combination of solutions corresponding to generalized eigenvectors.

If λ is a real eigenvalue, then every component of x(t) has the form e^λtP(t). In this case, P(t) is a real-valued polynomial.

If λ = α + iβ (i.e. if λ is a complex number) and its algebraic multiplicity q, then the eigenvector v will be complex too. Every component of x(t) has the form e^λtP(t), but this time P(t) is complex valued. If we write P(t) as Q(t) + i*R(t) and

 Then every component of x(t) has the form

In conclusion, every component of real solutions is the sum of functions that have the form

In this case, p and q are polynomials, and α is the real part of an eigenvalue.

Here’s a theorem for you:

“Let A be an n × n matrix.

1. Suppose that the real part of every eigenvalue is negative. Then every solution to the system x’ = Ax tends toward the equilibrium point at the origin at t → ∞.

2. Suppose that A has at least one eigenvalue with a positive real part. Then there are solutions to the system x’ = Ax starting arbitrarily close to the equilibrium point at the origin that get arbitrarily large as t → ∞” (430).

This theorem refers to both real and complex eigenvalues of A.

Suppose that y’ = f(y) is autonomous and y₀ is an equilibrium point. So we’ll say y­₀ is stable “if, for every ∈ > 0, there is a δ > 0 such that if y(t) is a solution that satisfies |y(0) – y₀| < δ, then y(0) – y₀| < ∈ for all t > 0” (431). In other words, y₀ is stable if every solution that starts close to y₀ stays close to y₀ as t increases.

The difference between an equilibrium point being stable and asymptotically stable is there is “an η > 0 such that every solution y(t) satisfying |y(0) – y₀| < η approaches y₀ as t increases” (431). In other words, every solution will stay reasonably close to y₀ if they start reasonably close to y₀.

Back in some other section, we talked about six different cases of curves. (http://differentialequationsjourney.blogspot.com/2013/10/93-last-really-long-summary-hopefully.html) A spiral sink and a nodal sink are both asymptotically stable. Therefore any asymptotically stable equilibrium point is called a sink.

There’s a distinction between stable and asymptotically stable because there are equilibrium points, such as a center, that are stable but not asymptotically stable.

An equilibrium point is unstable if “there is an ∈ > 0 such that for any δ > 0 there is a solution y(t) with |y(0) – y₀| < δ, but for which there are values of t with |y(0) – y₀| > ∈” (431). Examples of unstable equilibrium points are spiral sources, nodal sources, and saddle points. However, a saddle point isn’t source, which is the opposite of asymptotically stable.

So if we revise our theorem, for a linear system x’ = Ax has an asymptotically stable 0 if the real part of every eigenvalue of A is negative. But if there’s one eigenvalue with a positive real part, then 0 is unstable.

So that’s it for 9.7! Hooray!

Considering that 9.8 and 9.9 are fairly long, enjoy this while it lasts. I for one am not excited to summarize 9.8 and 9.9, but as of this summary, I have 17 more of these things to do before I’m free from the terror of summarize math things. Double hooray!

http://texas.math.ttu.edu/~gilliam/ttu/ode_pde_pdf/Ch4.pdf may or may not help you. I’m not very sure what a lemma is, but I’m sure it’s a magical device that can help us all understand math better. Also, I’m fairly confident this is talking about the same stuff we talked about today.