Math 119 A, Midterm


ODEs and Dynamical Systems

Solve the 2x2 ODE systems $\dot x = Ax$ by finding the eigenvalues and eigenvectors and exponentiating the matrix P^-1AP,

1.

$$A = {\left(\begin{array}{cc} 2 &6\\ 1 &-3 \end{array}\right)}.$$

2.

$$A = {\left(\begin{array}{cc} 1 &-2\\ 2 &1 \end{array}\right)}.$$

3.

$$A = {\left(\begin{array}{cc} 1 &4\\ -2 &2 \end{array}\right)}.$$

Draw the phase portraits for the three ODEs above and classify the flow as sinks, sources, centers, etc.

4.

$$A = {\left(\begin{array}{cc} 2 &6\\ 1 &-3 \end{array}\right)}.$$

5.

$$A = {\left(\begin{array}{cc} 1 &-2\\ 2 &1 \end{array}\right)}.$$

6.

$$A = {\left(\begin{array}{cc} 1 &4\\ -2 &2 \end{array}\right)}.$$

7.

Suppose that a matrix A has one complex eigenvalue $\lambda = a +i b$, with a < 0 and one real eigenvalue $\lambda = c < 0$ with (algebraic) multiplicity two but only one eigenvector. What does the normal form P^-1AP of the matrix look like and what is the solution y(t) of the ODE

$$\dot x = Ax$$

in the coordinates y = P^-1x ? What does the phase portrait look like?

8.

Find the subspaces E^s, E^c, E^u of the ODE

$$\dot x = {\left(\begin{array}{cccc} 1 &0 &0 &0 \\ 0 &0 &-1 &0 \\ 0 &1 &0 &0\\ 0 &0 &0 &-1 \end{array}\right)}x,$$

and find vectors x^s, x^c, x^u in each subspace. What happens to the solutions with these vectors as inital data as $t \to \infty$ ?