MATH 431 FALL 2005- COURSE LOG
8/24 Course outline and policies
Ex1:
1st order, linear, non-homog
Ex2:
x'=x^2- diagram of solutions, blowup in finite time
Autonomous equations: equilibria, phase line (intro)
READ: ch 1 of [1], sections 1.1, 1.2, 1.3
Problems: 2, 8, 9, 10 (ch.1)
8/26 Examples using dfield in MATLAB
Ex
1: a non-homog linear eqn with periodic steady-state
Ex2: autonomous eqn defined by a cubic polynomial
Ex3: logistic growth
3a:) constant harvesting- two regimes (small rate, large rate)
3b) time-dependent carrying capacity: a non-autonomous perturbation
Problems: 3, 4, 5,7 (ch.1)
8/29 Logistic equation: phase line
analysis, blowup in finite time of negative solutions
logistic with constant harvesting; stable vs. bifurcation values
of the parameter, bifurcation diagram
non-vanishing of derivative at equilibria and stable parameter values
Problem 13a: discussion
8/31 Problems3b
(bifurcation), 7a (discontinuities)
The local existence/uniqueness theorem: Picard's theorem
problems 11,
12a
9/2 Matlab class: based on [2], ch. 3 and
6
equilibria at critical points:: ch. 3 18, 19
sensitive dependence: ch.3 29, 31
periodic solutions (ex: logistic with periodic harvesting term)
(discontinuous coeffs: examples 4, 5, 6 in ch. 6- if time allows)
9/5 Labor day: no classes
9/7: The Poincare map and existence of periodic solutions
(handout with Matlab graphs)
(derivative of P at a fixed point vs. stability or instability of the
periodic soluttion)
Main example: logistic
equation with periodic harvesting term
Try this at home (or the computer lab):
FUN WITH
THE POINCARE MAP (Matlab)
9/9: Remarks on the Poincare map: always increasing, inherits
convexity/concavity properties from f(t,x); problems 15,16 (discussion)
intro to 2X2 linear
systems- chapter 3 in [1], basic use of pplane7- ch7 and pp. 217/219 in
[2]
(ch. 2 of [1] will be skipped, except for 2.6- read at home, especially
2.5)
Examples of phase plane diagrams
and (t,x), (t,y) graphs: the following vectors are (A,B,C,D) in
the Matlab input:
(2,-3,1,-2)=saddle
(1,-1,2,4)=unstable node (-1,0,1,-2)=
stable node (-1, -2, 8, -1)=stable
spiral (3 ,-2,5,-2)=unstable
spiral
(2,-4,2,-2)=center
(1,-1,4,-3)=degenerate focus (-4,2,2,-1)=very degenerate
Homework: obtain Matlab plots
of the phase plane and both (t,x) and (t,y) graphs for each example
(bring printouts to class!)
The systems on the first line have something in common. What is it?
(hint: change each coefficient by 0.01 and plot again;
then do the same for those on the second line.)
Homework set 1 (due 9/9 at the
start of class)- all problems from text [1], ch.1
2b, 2d clasification of equilibria (include Matlab plots with
representative solution curves)
3c- bifurcation diagram (include graphs of f(x) and of soln curves for
each range of the parameter a.)
13b,c: critical equilibria (justify your answers using Taylor series)
11c:- solution defined only for -1<t<1
10- Poincare map for a linear equation (this can be found "by hand",
after you compute the general solution)
15- Hint: Consider the
direction field;. If P is the Poincare map, show that P(p)>p and
P(q)<q.
16- Use Matlab to find the answer, then prove it. (Hint: find three values of p for
which f(t,p) has constant sign and use problem 15.)
9/12: autonomous linear 2X2 systems: vector fields, phase portraits,
connection with second order equations (ch.3 of [1])
9/14 (continued)
Friday 9/16: I'll be at a conference in California. Makeup: to be
scheduled (it will be on a Monday afternoon)
9/19 linear 2x2 systems: reduction to standard forms
9/21 continued: general solutions
Homework set 2: due Friday 9/23
at the start of class
Chapter 3 of [1]: 1 (justify), 2 (use Matlab for (d), only
original system needed), 9, 16 (use Matlab to help)
Chapter 4: 2, 3
9/23 stability and asymptotic stability of equilibria
conjugacy and
structural stability: informal definitions
hyperbolic linear
systems (2X2 case); hyperbolicity, conjugacy and str stablity in this
case
A "tour" of the
trace-determinant space (handout and Matlab) - identification of
non-hyperbolic and non-str stable systeems, and bifurcations
Example: general
solution when an eigenvalue is zero.
9/26: comments on HW2 problems
an example
showing the map that assigns to a 2X2 matrix its standard form is not
continuous
autonomous
nonlinear 2X2 systems- intro example (from ch.8 of [1]).
main idea:
consider the linearized system at the equilibria
9/28 Examples, cont'd: invariant sets and conserved quantities
9/30: examples of NL systems (Ch8 of [1]), using Matlab
Homework set 3: due Monday
10/10 at the start of class
p.185 of [1]:
1(ii),(iii), (iv);
5, 8, 9
p.232 of
[2]: 19, 32
Target date for first take-home
test: Wed 10/12, due Monday 10/17
Projects: start thinking about
your choice (each student a different one), let me know soon (first
come, first choice!)
all involve some use of Matlab- in addition to the outline given in the
text, the original papers should be consulted
p. 230: chemical reactions that oscillate (also 9 on p.137 of [2])-
MARZOUK
p.252: competition and harvesting (math ecology)- O'CONNOR
p.272: neurodynamics(Fitzhugh-Nagumo system)-CARR
p.324: the Roessler attractor (also p.138 of [2])-BEACHBOARD
p.298: classical limits of quantum mechanical systems-KOOP
p. 297: non-Newtonian central force problems-DAWSON
p.129 of [2]: motion of spacecraft near Lagrange points-ASHWORTH
p.114 ff and p.129 of 2 and ch. 14 of [1]: the Lorenz system (climate
modelling)-THERRIEN
(If none of these seems appealing, you'll need to come up with a
nonlinear system of
ODE in your field exhibiting interesting behavior, occurring in the
recent research literature!)
10/3: examples with non-hyperbolic equilibria: saddle-node bifurcation
10/5: Stability via Liapunov's method
10/7 (Matlab) Examples of global analysis of 2D autonomous
systems and bifurcations:
saddle-node
bifurcation with invariant unit circle
Remark 1: this example has 2
bifurcation values, a=-1 (problem 7 in ch.8; done in class) and a=0.
Continue the Matlab study for a=0, using the discussion in the text as
a guide. Think about the following question:
both bifurcations are of saddle-node type. Are the same pairs
saddle/node collapsing in both cases, or is
there an `exchange' at a=0?
Remark 2: now that all the project
topics have been assigned (see above), you should be tracking down the
scientific
article(s) describing the origin of the model (a reference is given in
each case) and starting to read it(them). Please
make
me a copy of the article if you'd like me to help!
10/10 (morning): Liapunov functions (cont'd) -HW 3 due
comments on HW3, problem 9
10/10 (5:00-6:00 pm- makeup class)
Matlab (in computer lab)- further examples: saddle connection, pendulum
handout given:
systems in polar coordinates
Remark on the handout: the
linearization computed on page 2 is true everywhere, except at the
origin.
At the origin,
if f(x,y) and g(x,y) are both differentiable, with differentials (a,b)
(resp. (c,d)), tht linearization is
given by the 2X2
matrix with 1st row (a,b), 2nd row (c,d).
10/12: Take-home Exam 1 given.
(Individual, due
10/17)
Exam 1
Pendulum, with and without damping: conderved quantity, Liapunov
function.
LaSalle's invariance principle.
Definition of alpha- and omega- limit sets
10/14 FALL BREAK (no class)
10/17 gradient flows
10/19 Hamiltonian systems
HW set 4- due Monday, 10/31
Chapter 9- 6, 8b,c
12, 13, 16
10/21 class canceled- makeup TBA (on a
Monday afternoon)
10/24 Limit sets and periodic orbits
(autonomous systems in 2D)
Definition, examples of alpha and omega-limit sets; local
sections and flow boxes;
existence of a continuous `hitting time'- proof using IFT.
10/26 The Poincare map for
two-dimensional autonomous systems
10/28 Examples: Lagrangian and
Hamiltonian systems, graphs of separatices as limit sets (Matlab)
10/31 The Poincare-Bendixson
theorem; main steps in the proof
11/2 Poincare-Bendixson
theorem- examples, applications
HW set 5-due Monday, 11/14
Chapter 10- 1, 2, 8, 9, 12, 16
You may use the following fact (easy to show): an area-preserving
system cannot have limit cycles
11/4
Bendixson's criterion
Competing species, predator-prey systems- examples (Matlab)
EXAMPLES
IN POPULATION BIOLOGY
11/7
The van der Pol system: definition of the Poincare map
11/9 V. der Pol: stability of the periodic orbit; Lienard's theorem.
DUE
DATE FOR THE PROJECT: 11/30 PROJECT
DETAILS
2nd
take-home exam: given out 11/18, due 11/23
Makeup class: Monday 11/14, topic: index of singular points of planar
vector fields.
11/11 Discussion of HW
problems.
Further hint for no. 2: the Poincare map is defined on a section
{(x,z); x>0} by P(x,z)=(x(2pi), z(2pi)), where ( x(t) , y(t), z(t))
is the solution with
initial data (x,0,z) (or in cylindrical coordinates: r=x, theta=0, z),
which is easily computed explicitly.
PLANAR
AUTONOMOUS SYSTEMS WITH PERIODIC SOLUTIONS
11/14 Morning: Autonomous systems in
mechanics
Afternoon (makeup class): Poincare index of vector fields (via line
integrals)
11/16
Central force fields: angular momentum, reduction of the phase space,
eqns of motion in polar coordinates
11/18
Poincare index of vector fields (handout)
Handout- slns of HW set 5
Take-home Exam2
given out
11/21
The Newtonian central-force system
11/23
Kepler's first law (solution of the Newtonian system)
Take-home exam 2 due
11/25
Thanksgiving holiday (no classes)
11/28
Solution of exam 2
11/30 Project
presentations: Belousov-Zhabotinsky (Marzouk), Lorenz system
(Therrien),
competing species with harvesting (O'Connor)
12/2
Project presentations: Fitzhugh-Nagumo (Carr), Roessler attractor
(Beachboard)
12/5
Project presentations: collision surfaces for non-Newtonian central
force (Dawson),
pendulum with constante torque (Koop)
12/9
FINAL EXAM- Friday, 12/9,
10:15-12:15 (in the usual classroom)
open book- you may use the text, class notes and handouts
TOPICS INCLUDED: ch. 9, 10, 12, 13 (only sections introduced in class)
handout on the index of vector fields;
REVIEW: 1) HW problems for ch. 9 and 10; 2) Exam 2
3) HW problems for ch. 13: 1, 8, 9, 10
(If
there is time, you may also wish to look at the Matlab examples from
11/4 and 11/11-
see links above)
Final exam
(PDF)