Finish Chapter 3

Due Monday Jan 12.

Ancient History

Due Weds Jan 21 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

Due Mon Jan 26 in class.

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.

Due Mon Jan 26 or Weds Jan 28 in class.

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. )

.

Due Mon Jan 33 = Mon Feb 2 in class.

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)

.

Due Mon Feb 9 in class.

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.

2. Solve numerically the few simple ODE I wrote
on the board in class.
See Mma Help:

1.6.4 Numerical Differential Equations

9.4.2

9.5.5 EXCLUDE a

9.5.5 b c

9.6.2

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.

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

Friday Feb 13 Homework Quiz

.

Due Weds Feb 25 in class.

3.4.13

3.4.14 Consider dx/dt= r x + x^3 -x^5
which exhibits a subcritical bifurcation,

(a) Find an algebraic expression for all fixed points as varies.

Do (b)(c)

3.4.15

Let Y be a symmetric matrix.

Define the quadratic function f: R^n to R by

f(x):= Yx.x

G1. What is the gradient of f? Note at each vector x the gradient
is a row vector; which row vector is it?

.

The partial differential inequality

Ax . gradient f(x) <0
for all x

is called a Lyapunov inequality.

G2. Express it in terms of the matrices A and Y.

Due Weds Feb 25+7 in class.

1. Express
q(s,t):= s t + 4s^2 + 3t^2

in the form q(s,t)= M v.v where M ia a 2x2 symmetric matrix
and v= (s,t)^T.

2. Express
Yx . (Ax + Bu) + 1/2 [ u^2 - Cx . Cx ] >0
for all x and u

in the form M v.v where M ia a 2x2 symmetric matrix
with matrix entries.
Here
and v is the column vector
with entries (x,u).

.

3. Do the computer HW problem assigned in class on Weds Feb 27.
It was:

Generate an output function $y(t)$
from system

dx/dt = f(x)
y = C x

where
f(x)= Ax and C were given in class.
(The matrices A and C were

A= 1 3

2 1.5

and
C= (1 , 5).
)

3a Compare the delay construction v^2(t) to the
corresponding trajectories of the differential equation dx/dt= A x.

3b Fiddle a lot.

3c Make up a nonlinear f and try the same thing.
.

Due Weds Feb 25+7+7 = Mar 10 in class.

1. \ Compare trajectories of the system given in class with the
ParametricPlot3D of the v^3(t).
What is extremely distinctive about this parametric plot?

2. \ Demonstrate how you can use Mma or Matlab to see if 3 given vectors in R^4
are linearly dependent? Nearly linearly dependent?
How do you determine the dimension of the subspace of R^4 which they span?

(Hint: the rank of a matrix 4 x 3 matrix equals its column rank
equals the dimension of the space spanned by the columns of M. )

(Hint: The rank of M equals the number of nonzero eigenvalues of MM^T.)

3. \
Suppose

y(t)= e^{3t} + e^{-t}sin{4t} + e{2t}

is the output of a system

dx/dt= f(x) y= g(x)

starting at some initial condition which we do not know.
What is the dimension of the statespace of the system.

4. Extra credit: \
Is there a linear system which generates this data?
How do you tell? Im not sure I know a great solution,
this is mostly for your enjoyment.

5. DISSIPATION:
Consider the linear system with one dimensional state space:

dx/dt= ax + u y= (0.1)x

For which choices of the parameter a is the system dissipative?

6. Dynamic programing question:
The required supply function S_r for a reachable
system is finite at all points in the statespace.
Prove it is a storage function, that is,
it satisfies the storage function inequality.