HW 1

1. Apply the energy balance method to each to the following equations. Find the
amplitude and stability of any limit cycle for small $\epsilon$.

(a) x'' + \epsilon (x^2+(x')^2 -1)x' + x = 0,
(b) x'' + \epsilon (1/3 (x')^3 - x') + x = 0.
 
Hand in one for grading.

2. Other examples, do not hand in
x'' + e(x^4-1) x' + x =0,
x'' + e(|x| -1) x' + x =0,
x'' + e(x'-3)(x'+1)x' + x =0,
x'' + e(x-3)(x+1)x' + x =0.

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HW 2

1. Plot the Lorenz system 
---cut here lorenz.ode---
# the famous Lorenz equation set up for animated waterwheel and 
# some delayed coordinates as well 
init x=-7.5  y=-3.6  z=30 
par r=27  s=10  b=2.66666 
par c=.2  del=.1 
x'=s*(-x+y) 
y'=r*x-y-x*z 
z'=-b*z+x*y 
# x is proportional to the angular velocity so integral is angle 
theta'=c*x 
th[0..7]=theta+2*pi*[j]/8 
# approximate the velocity vector in the butterfly coords 
z1=z-del*(-b*z+x*y) 
x1=x-del*(s*(-x+y)) 
@ dt=.025, total=40, xplot=x,yplot=y,zplot=z,axes=3d 
@ xmin=-20,xmax=20,ymin=-30,ymax=30,zmin=0,zmax=50 
@ xlo=-1.5,ylo=-2,xhi=1.5,yhi=2,bound=10000 
done 

2. Optional: Give an example of a two dimensional system which has an orbit
whose \omega limit set is not empty and is disconnected.

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HW 3

(1) Page 210: 4 or 6, hand in one only
  
optional: Consider x' - x = f(t). 
If f is continuous and |f(t)| < C for all t,
show that there exists a unique bounded solution x(t) defined for all t.
This problem is an example of Fredholm's alternative and will be generalized
greatly: 
In the space of bounded
functions, the homogeneous system
x' - x = 0 
has no nonzero solution, thus the nonhomogeneous system x' - x = f
has a unique bounded solution for all bounded function f(t).

(2) Page 219: 1

(3) Let P(s) be the Poincare map for a focus at the origin of a planar system
with b \neq 0. If d(0)= d'(0)= \dots = d^{k-1}(0) and d^k(0) \neq 0.
Show that k is odd. 
Hint: show that d(s).d(-s) < 0 first. This comes from the
physical interpretation of what r<0 means in polar coordinates.

(\neq means not equal.)

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HW 4

(1) Hand in: 
Consider x'' = f(t) where f is continuous and periodic with period T.
Give a direct proof that the equation has a periodic solution of period T if and
only if 
\int_0^T f(t) dt = 0.
Obtain an explicit formula for all the T-periodic solution of the equation.

(2) Optional: Show that if g(t) is not orthogonal to all the periodic solutions
of the adjoint equation with period T, then all the solutions of
x' = A(t)x + g(t) are unbounded.

(3) optional: Consider the periodic system x'=A(t)x, A(t+T)=A(t).
Let \Phi(t) be a F.M.S. 
Let \Psi(t,s) = \Phi(t)\Phi^{-1}(s) be the Pricipal Matrix Solution (Transition
matrix).
Show that the characteristic multipliers can be defined as the eigenvalues of 
the matrix \Psi(T+s,s) for any real number s. (Usually they are defined as the
eigehvalues of \Psi(T,0).)

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HW 5
 
Page 231: 1,2,5,  hand in one.
page 251: 1,2  do not hand in.
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HW 6

page 341: 1 and 2, hand in one.
page 348: 2

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HW 7

page 359: 1 (a) or (b) hand in one only.
page 382: 7.

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HW8
PAGE 430: 3.

Prove directly that if A is a 2x2 matrix, u and v are 2 dimensional
vectors,
then
Au wedge v + u wedge Av = trace (A) times (u wedge v)
How to generalize this to R^n?
Keep the proof to yourself for future reference. Do not hand in.