9.3, the last really long summary (hopefully)

Section 9.3 is entitled “Phase Plane Portraits.”

Let us recall what we know about the system y’ = Ay.

Notice that this system is autonomous, which means we’ll be looking at the phase plane to visualize solutions. If we had a linear homogeneous system Ay = 0, then our set of equilibrium points would be v ∈ R² such that Av = 0. In other words, the set is the nullspace of A. If A is nonsingular, then the only equilibrium point is the origin 0. If A is singular, the equilibrium points are the points in its nullspace (which could be a line or all of R²).

In any event, these cases have special names. The book goes over 6 important cases. Three of these cases have real eigenvalues, and the other three have complex eigenvalues.

The six types (http://staff.www.ltu.se)

Real Eigenvalues

So when we’re considering real eigenvalues of A, recall that we know T² – 4D > 0. This means our eigenvalues are

Something to note is that λ₁ < λ₂.

Also from last section, we know that the general solution is

Like last time, C₁ and C₂ are arbitrary constants and v₁ and v₂ are eigenvectors that are associated with λ₁ and λ₂.

If either C₁ or C₂ is equal to zero, then there are the solutions

These are called exponential solutions. If we simplified things down to λ being an eigenvalue for A and v being the associated eigenvector, then exponential solutions have the form

Something to note about this form is that y(t) will always be a multiple of v. The solution traces out a half-line consisting of positive multiples of Cv. There are two solution curves in the phase plane that depend on the sign of C. Since these solutions are half-lines, exponential solutions are also known as half-line solutions.

There are two cases when you consider λ: when λ > 0 and when λ < 0. When λ > 0, e^λt increases. These solutions go away from the equilibrium point at the origin when t increases and goes to the origin as t gets negative. These type of solutions are called unstable solutions. Therefore, when there is a positive eigenvalue, there are two unstable half-line solution curves.

The other case is when λ < 0. Not surprisingly, this is the exact opposite (e^λt decreases, go away from equilibrium point at the origin when t decreases and approaches the origin when t increases) so these solutions are called stable solutions. This means for negative eigenvalues there are two stable half-line solution curves.

Case one: saddle point

The first case is when eigenvalues are real and have different signs, so λ₁ < 0 < λ₂.

When C₂ = 0, we will get two stable half-line solutions. When C₁ = 0, we will get two unstable half-line solutions.

This would be a superposition of the two exponential solutions. As t gets very large, the first part of the solution (with C₁) will tend to 0 so the solution as a whole will tend to the latter part (with C₂). The opposite will happen when t gets very negative. 

“Geometrically, this means that the solution curve goes to ∞, asymptotic to the half-line generated by C₁v₁” (394).

Case two: nodal sink

The next case is when both eigenvalues are negative, i.e. λ₁ < λ₂ < 0. The solution is still

Like last time, if either of the constants are zero, the solution is exponential. This time, however, both eigenvalues are negative so all of the solutions are stable. When both C₁ and C₂ are nonzero, we get the superposition of both exponential solutions:

In the first rewriting, the bracketed part converges to C­₂v₂. In the second rewriting, the bracketed part approaches C₁v₁.

Something cool to note about linear systems with two negative eigenvalues is that “…all solution curves approach the origin as t → ∞ with a well-defined tangent line” (397). Solutions with this type of behavior (approaching their equilibrium points as t goes to infinity) are called nodal sinks. If the solution curves approach their equilibrium points as t goes to negative infinity with a well-defined tangent line, they are called nodal sources. Thus planar systems with two negative eigenvalues will have a nodal sink at the origin.

Case three: nodal source

This case is where both eigenvalues are positive, i.e. 0 < λ₁ < λ₂. We already defined what a nodal source is. In other words, a nodal source parallels a nodal sink, but with time reversed.

Complex Eigenvalues

So the final three cases are when the eigenvalues are complex.

Case one: center

This is the case where the eigenvalues are purely imaginary. This means that α = 0, so then the general solution will become

The trigonometric functions are both periodic. They will have a period 2π/|β|. This means that y(t) will have the same property. This means that the solution trajectory is a closed curve since it orbits the origin with its period.

The distinguishing characteristic of this case is that the equilibrium point is surrounded by closed solution curves. This property is why the equilibrium points for planar systems are call centers. This means that planar systems with purely imaginary eigenvalues will have centers at the origin.

Case two: spiral sink

This is the case where the real part of the eigenvalue is negative. The general solution would look like this:

The term inside of the brackets is what we looked at in the center case. These terms are periodic with a period T = 2π/|β|. By themselves, they parameterize ellipses centered at the origin. However, they are now being modified with the factor of exp(αt). This was the case where α is negative, so that exponential will go to zero as t gets very negative. This means that there are solution curves circles the origin, but these curves are also drawn toward the origin, which results in a spiral motion. Because the solution curves spiral to the equilibrium point, they are all stable. This characteristic makes the solutions spiral sinks. Therefore any linear systems with complex eigenvalues having negative real parts have a spiral sink at the origin.

Case three: spiral source

This is the case where the real part of the eigenvalue is positive. The general solution has a form very similar to the spiral sink, but with α being positive. This means the amplitude of oscillation will increase as solutions spiral about the origin. This is the behavior of a spiral source. Therefore, linear systems with complex eigenvalues with positive real parts have spiral source at the origin.

Sometimes you want to know whether the motion of a spiral is clockwise or counterclockwise, but you have no Google or calculator to tell you the answer. One way you can find out is to compute a vector in the vector field determined by the right hand side of your system.

If we have a matrix

We also have complex eigenvalues, so we can compute the vector field at, say, (1, 0)^T.

If a₂₁ > 0, then the vector will point into the upper half-plane, which means the rotation is counterclockwise. If a₂₁ < 0, then the vector will point into the lower half plane, meaning that the rotation is clockwise.

That’s all for 9.3! It was a lot to take in. It was also a lot to write out. Yay content-heavy sections!

I’ll see you when I see you.