Math 426 Final: Due Wed. Dec 17 before
2:00P.M. (No Extensions!)
December 16, 2003
1. The system x = x y + 2 sin(x), y = x + y + x2 y has a rest point at
the origin. Draw (with explanation) the phase portrait of this system
near the origin. Det
Math 426 Homework 1
September 7, 2003
Hand in problems 5,7,9,13 on Sept. 1.
Prepare the rest of the problems to present to the class starting Aug. 27.
General rules: Please do not consult books where you can nd the answers
to exercises. Also, d
Math 426 Homework 4
September 21, 2003
1. Prove that the phase portrait of the 2nd-order ODE x x + x2 = 0 has
a homoclinic orbit.
2. Let H (x, y, z ) = xyz . Does XH := grad H 3H , where denotes the
outer unit normal on the unit sphere, have an
Math 426 Homework 3
September 23, 2003
1. Exercise 1.34 p. 26 of the book. Consider a Newtonian particle of
mass m moving under the inuence of the potential U . If the position
coordinate is denoted by
q = (q1 , . . . , qn ),
then the equation
Math 426 Homework 2
September 4, 2003
1. Construct innitely many dierent solutions of the initial value problem
x = x 1/3 ,
x(0) = 0.
Why does the Existence and Uniqueness Theorem for dierential equations fail to apply in this case?
2. Prove th
Math 426 Homework 5
October 12, 2003
1. Draw the phase portrait of Newtonian system x = x x3 . Give a qual
itative description of solution of the system for the initial condition
(x(0), x(0) = (1, 1/ 2). How would the motion change qualitat
Math 426 Homework 7
November 4, 2003
1. Prove that the system
x = x y x3 ,
y = x + y y3
has a unique globally attracting limit cycle on the punctured plane;
that is, the plane with the origin removed.
2. Can a system of the form
x = y,
y = f (x
Math 426 Homework 6
October 21, 2003
1. Find an explicit formula for the ow of the dierential equaton
x = y + x(1 x2 y 2 ),
y = x + y (1 x2 y 2 ).
Show that all orbits except the rest point at the origin are asymptotic
to a limit cycle.
Math 426 Homework 8
October 27, 2002
2. The linearized Hills equations for the relative motion of two satellites
with respect to a circular reference orbit about the earth are given by
x 2ny 3n2 x = 0,
y + 2nx = 0,
z+n z = 0
where n is
Math 426 Homework 10
December 9, 2002
1. Find the principal fundamental matrix solution at t = 0 for the MarkusYamabe system: x = A(t)x, where A(t) is given on page 171. Write
the principal fundamental matrix solution in real Floquet normal for