ODEs and Dynamical Systems
1. (a) 5 pts. Solve the 2x2 ODE system _x = Ax by nding the eigenval-
ues and eigenvectors and exponentiating the matrix P 1AP,
(b) 5 pts. Draw the phase portraits for the ODE above and classify
the ow as a sink, source, center, etc.
2. 10 pts. Solve the 3x3 ODE system _x = Ax by nding the eigenvalues
and eigenvectors and exponentiating the matrix P 1AP,
3. (15 points total) Do the qualitative analysis of the equation,
x+ cos(x) = 0;
(a) (3 points) Find the stationary solutions.
(b) (3 points) Determine the stability of the stationary solutions.
(c) (3 points) Draw the phase portrait of the equation.
(d) (3 points) Use the phase portrait and (a) and (b) to identify four
di erent types of solutions of the equation and describe their qual-
itative behavior.
(e) (3 points) How would you go about computing the homoclinic
orbits?
4. (10 points) Find the (linear) stable and unstable manifolds of the sta-
tionary solution of the system
5. (10 points) Does the IVP:
_x = x1=4; x(0) = 0;
have a unique solutions? How many solutions does it have? Justify
your answer.
6. (10 points) Show that the equation
x+ cos(x) = 0;
has global solutions that exist for all time.
7. (15 points) Show that the origin is a stable solution of the equation
x+ _x+x x3 = 0:
(10 points) Then nd the energy of the equation
x+x x3 = 0;
and use it to show that for this second equation the origin is (Lyapunov)
stable.
8. (15 points, total) Use the Poincar e map to describe the solutions of
x+ _x+ cos(x) = cos(!t):
In particular
(a) (5 points) Draw the phase portrait.
(b) (5 points) Describe the trajectories close to the origin.
(c) (5 points) Describe what happens to the homoclinic orbits.