Office hours:MW 5-6 and by appointment
Office: APM 5256, tel. 534-2734
Teaching assistant: Orest Bucicovschi, office: APM 6434, office hours: email: firstname.lastname@example.org
Computation of grade: homework: 20%, midterm: 30%, final 50%.
Dates of exams: No make-up exams!
Final: Wednesday, March 21, 3-6
Course material: We will use the book "Differential Equations, Dynamical Systems, and an Introduction to Chaos", by Hirsh, Smale and Devaney.
Homework assignments Unless further notice, homework is to be turned in BY NOON on or before the date posted in section. There will be a drop box on the 6th floor of APM, marked for our course (turn right leaving the elevator).
I will try to get the homework assignment on the net in time.
Due to time and other limitations, this may not always be possible.
The fact that there is no assignment posted for a particular date does
therefore NOT necessarily mean that no homework is due.
for 1/19: p. 231: 1bc, 2, 6ab
for 1/26: p. 273: 1, p233: 6c (hints: 1) First show that the system x'= y, y'=-x^3+x has solution curves of the form H(x,y)=c, where c is a constant and H(x,y)=x^4/4 - x^2/2 +y^2/2; see also p.209/210 and recall properties of Hamiltonian systems. 2) determine in which direction the tangent vectors of solutions of the system given in 6 show on the curves of the form H(x,y)=c. Use this to determine the limit sets).
for 2/2: 1) Consider x'=x-y-x(x^2+5y^2), y'= x+y - y(x^2+y^2). a) Classify the fixed point at the origin. b) Rewrite the system in polar coordiantes, using rr'=xx' + yy' and \theta'=(xy'-yx')/r^2. c) Determine the circle of maximum radius r_1, centered at the origin such that all trajectories have a radially outward component in it. d) Determine the circle of minimum radius r_2, centered at the origin such that all trajectories have a radially inward component in it. e) Prove that the system has a limit cycle somewhere in the region r_1 lessequal r lessequal r_2.
2) Show that the system x'=y-x^3, y'= - x-y^3 has no closed orbits by constructing a Liapunov function V=ax^2+by^2 for suitable a and b.
for 2/9: p. 325: 6; Problem 1 on page 273 was way too hard without any hints. Try again using the following: Use software atto formulate conjectures about what the phase portrait should look like (e.g. how many periodic solutions, symmetries etc). You should do that before looking at the hints below (finding conjectures is the fun part!).
Phase portrait software
Hints for proving some of your conjectures: (1) Show that the tangent lines of solutions are symmetric with respect to the y-axis. (2) Show that the solutions themselves are symmetric with respect to the y-axis (Hint: assume X(t)=(x(t), y(t)) is a local solution for t in the interval [-a,a] for some suitable a > 0; show that we also have a local solution of the form (-x(-t), y(-t)) for t in the interval [-a,a]). (3) Show that any solution starting at the positive y-axis will hit the negative y-axis and will belong to a periodic solution (see the proof on p. 264, which we also did in class). (4) Find constants a, b, c such that x(t)=at, y(t)=bt^2+c is a solution.
for 2/16: p. 325: 5 (recall that any solution through a point (x,0,27) will tend to the origin) and Show that for given y coordinate y=y_0 there is at most one point (x,y_0) in the rectangle R (as in the book) which is periodic, i.e. for which there exists an n such that \Phi^n(x,y_0)=(x,y_0). and Sketch the regions \Phi^2(R), \Phi^3(R), \Phi^4(R)... Where would you expect to lie most of the attractor A?
for 2/23: p. 354: 1c, 2d, 4, 8 (see pictures on page 335 and read the discussion before that).
for 3/2: p. 355: 5, 11
for 3/9: p. 356: 16 (you need not worry about proving continuity of the conjuagacy map), 17 and: Show that the periodic points in the discrete dynamical system (\Sigma, \sigma) of sequence of 0's and 1's with shift \sigma are dense in \Sigma.
for 3/16: p. 300: 7 (or 6): just do for one of these systems what has been done for the Newtonian force field in 13.4. Do you also get a torus?
For further information and some practice exams, you may consult the following webpage of the same course given by a colleague of mine, based on a previous version of our book.