[MUSIC] So we've been solving this
differential equation Ẋ = Ax. A is a two-by-two matrix. X is a column vector X1 and X2. In the next series of lectures, I want to show you how to visualize
the solution of this equation. Those diagrams are called
phase portraits and the visualization is done in what's
called the phase space of the solution. So X is a column vector X1 and X2 and the visualization then draws the solution of X2 versus X1. So first we have to say something
about this solution, the origin here. The 00 value is called a fixed point. So that's the point right at the origin. The meaning of a fixed point is that if the solution starts at 00. So if the initial condition for
X was 00 then it stays at 00 for all time. So the solution is fixed at 00
because the derivative is 0. So the origin is considered a fixed point. It's also sometimes called
an equilibrium point of the equation and the equilibrium point can be stable or
unstable. So if an equilibrium point is stable, it means that all nearby solutions
will be attracted to the equilibrium. But if an equilibrium point is unstable, it means all solutions will
run away from the equilibrium. Because this is a linear equation, it doesn't matter whether you're near or
far from the equilibrium. If this fixed point or equilibrium is stable all solutions
will then converge onto the fixed point. Unstable, they will all run away. So in the next few lectures, I want to classify
the behavior of this diagram. This diagram we solve by putting some
initial condition here so say over here. So this would be the value
X1 at T equals 0 and X2 at T equals 0 and then follow what the trajectory of this
initial point will be okay? So that means we solve
the differential equation and we follow where this point
moves in the phase space. And we will then draw diagrams for
several initial conditions surrounding this fixed point and
see what the diagram looks like. Okay, in this introductory
video I want to try and classify what the different solutions will
look like so let me make a table here. So the solutions or the qualitative
nature of the solutions will depend on the eigenvalues and
the eigenvectors of the matrix A. So if we just concentrate
on the eigenvalues, I can make a table of eigenvalues. And then fixed point. How does the fixed point behave? Okay, so
what could the eigenvalues look like? They can be real, so
let's say real eigenvalue so to real distinct eigenvalues and
they can both be negative. Okay, so two negative real eigenvalues. Then we know that the solution
has an exponential behavior. And because the eigenvalue is negative,
it's a decaying exponential. So in this case all the trajectories
will converge onto the origin. We call this fixed point a stable node okay? The other case would be real, and
both the eigenvalues are positive. Then the exponentials are growing in time. Everything's running away from the origin. This is called an unstable node. Okay, the other case then is one
positive and one negative eigenvalue. This is called a saddle point. Okay, the fixed point here
is called a saddle point. We'll see what that means is that
along one of the eigenvectors the solution is going into the origin and
along another eigenvector. The solution is running
away from the origin. Okay, apart from real eigenvalues,
we can have complex eigenvalues. Okay, and then there are two cases here. They're all called spiral points, but what matters here is
the sign of the real part. So if the real part of the complex
eigenvalue is negative, then this is called a stable spiral. And the solution then
spirals into the origin. If the real part is positive,
this is called an unstable spiral. Okay, so that's going to be our
classification of the fixed point. And the diagram the phase portrait for these different cases will all
have a distinctive look for them. So let me review. We're looking for
a picture that represents the solution of our system
of first-order equations. That picture is called
a phase portrait and the character of the picture
will depend on the type of eigenvalues that we get from the matrix A. I'm Jeff Chasnoff. Thanks for watching, and
I'll see you in the next video.