Finish Chapter 3
Due Monday Jan 12.
Ancient History
Due Weds Jan 21 in class.
Due Mon Jan 26 in class.
Due Mon Jan 26 or Weds Jan 28 in class.
.
Due Mon Jan 33 = Mon Feb 2 in class.
.
Due Mon Feb 9 in class.
2. Solve numerically the few simple ODE I wrote
on the board in class.
See Mma Help:
9.4.2
Some of you might like to take a look at this
article which is about 2 years old
Latest Lyapunov Technique
This gives details of the Lyapunov function method
I mentioned in class.
It would be tough reading but will give you an idea
of what a research article looks like.
They take a long time to read.
.
Due Mon Feb 16 in class.
Ch 3     3.40, 3.42(a)(b), 3.44 (Hint: rule out cycles in
certain regions - then rule out crosssing between the region.)
Ch 4     4.2,
4.5 EXCLUDE spiral saddle part of question
4.7
Ch 3
   
3.37, 3.38 Poincare map
Figure 4.5 in AP looks wrong. Is it?
Give a crude picture describing the Poincare map
if you do not believe the books picture.
Newtons method for finding the 0's of a function f goes as follows.
Define
g(x):= x - [ f'(   x   )]^(-1) f(  x   )
where f'(x) is the linearization of f at x,
and we assume it is invertible and denote its inverse by
   
   
   
[ f'(   x   )]^(-1)
Newton's method is the iteration
x(n+1)= g( x(n) )
Note a fixed point x* of g, at which f'(x*) is an invertible matrix,
satisfies f( x*)=0.
a. For one dimensional x.
Show that such a fixed point is always a stable fixed point
of g. ( If it were not nobody would use Netwons method.)
(Hint: linearize g at x*)
b. EXCLUDED Extra credit. For any dimensional x.
Show that such a fixed point is always a stable fixed point
of g. ( If it were not nobody would use Netwons method.)
(Hint: linearize g at x*; you may have too much difficulty,
but at least think about what is involved. )
Ch 4     4.16 Poincare map
4.19 Terminology
.
Ch 9 Stroglatz    
sec 9.2     you may want to use Mma or Matlab.
9.2.1 9 (b)(c),   9.2.2,
  9.2. 4,
  9.2. 6 (a)(b)
1. Suppose that f and g map R^n into R^n.
Let Dg(x*) and Df(g(x*)) denote the linearizations
of g at x* and f at g(x*).
(a) State the chain rule for h(x)= f(g(x)) at x* when n=1.
(b) Prove it when n=1 ( fond memories of M20A).
(c) What do you guess the chain rule says for arbitray n.
(d) EXTRA CREDIT. Prove it for arbitray n.
    1.6.4 Numerical Differential Equations
9.5.5 EXCLUDE a
9.5.5 b c
9.6.2
9.3.2 thru 9.3.7 or pick some of your own parameters and do
your own study.
9.6.5
3.1.5, 3.1.5(a)
3.2.2