If a coupled system of differential equations involves strictly linear equations of x and y, i.e. is of the form
for real numbers a,b,c, and d, then the system can be written in matrix form as
If M has real and distinct eigenvalues λ1 and λ2 that produce the corresponding eigenvectors p1 and p2, the general solution of the system is given by
If as t increases the solution curves move
all towards the equilibrium point, then the point is stable
all away from the equilibrium point, then the point is unstable
some towards and some away from the equilibrium point, then the point is a saddle point
For a coupled system whose matrix M has real and distinct eigenvalues, the phase portrait will have stream lines that approach the direction of the eigenvector with the larger eigenvalue. The process for sketching is as follows:
Draw the eigenvectors as straight lines through the origin (or other equilibrium point)
Consider which case the eigenvalues satisfy:
The sign of the eigenvalues determines whether we have an unstable node, stable node or saddle point.
The solution curves start in the direction of v2, the eigenvector with the smaller eigenvalue, and end in the direction of v1 with the larger eigenvalue.
Draw at least one solution curve in each quadrant generated by the intersecting eigenvectors.
When the eigenvalues of a coupled system are complex, say
the solution curves follow 3 possible shapes in a phase portrait:
a>0: an unstable spiral,
a=0: an ellipse,
a<0: a stable spiral.
Once the shape is known, we can determine whether the trajectories are clockwise or counterclockwise by evaluating dtdx at (0,1):
dtdx<0, spiral is counterclockwise
dtdx<0. spiral is clockwise
This is equivalent to looking at the top right of the matrix:
Nice work completing Linear systems and phase portraits, here's a quick recap of what we covered:
Exercises checked off
If a coupled system of differential equations involves strictly linear equations of x and y, i.e. is of the form
for real numbers a,b,c, and d, then the system can be written in matrix form as
If M has real and distinct eigenvalues λ1 and λ2 that produce the corresponding eigenvectors p1 and p2, the general solution of the system is given by
If as t increases the solution curves move
all towards the equilibrium point, then the point is stable
all away from the equilibrium point, then the point is unstable
some towards and some away from the equilibrium point, then the point is a saddle point
For a coupled system whose matrix M has real and distinct eigenvalues, the phase portrait will have stream lines that approach the direction of the eigenvector with the larger eigenvalue. The process for sketching is as follows:
Draw the eigenvectors as straight lines through the origin (or other equilibrium point)
Consider which case the eigenvalues satisfy:
The sign of the eigenvalues determines whether we have an unstable node, stable node or saddle point.
The solution curves start in the direction of v2, the eigenvector with the smaller eigenvalue, and end in the direction of v1 with the larger eigenvalue.
Draw at least one solution curve in each quadrant generated by the intersecting eigenvectors.
When the eigenvalues of a coupled system are complex, say
the solution curves follow 3 possible shapes in a phase portrait:
a>0: an unstable spiral,
a=0: an ellipse,
a<0: a stable spiral.
Once the shape is known, we can determine whether the trajectories are clockwise or counterclockwise by evaluating dtdx at (0,1):
dtdx<0, spiral is counterclockwise
dtdx<0. spiral is clockwise
This is equivalent to looking at the top right of the matrix:
Nice work completing Linear systems and phase portraits, here's a quick recap of what we covered:
Exercises checked off
On IB exams, you will not be expected to:
On IB exams, you will not be expected to: