Bruce B. Peckham - Fall 2011 Dynamical Systems (Math 5260)

Math 5260 Dynamical Systems

Fall 2011 Course Homepage
Prof. Bruce Peckham, Department of Mathematics and Statistics, University of Minnesota, Duluth

Syllabus
Dynamical Systems software links (Not all are fully functional. Caution: Some links lead to commercial sites, none of which are endorsed by the instructor. :) )
Check grades recorded to date.

Homework Assignments

Set 1, due FRI 9/16
1. Strogatz: Read Preface and Chapter 1: Overview.
2. Devaney: Read Preface and Ch's 1-4
3. Ch 3: 1,3,4,5,6,7abh,8,11,12,13,14
4. Ch 4: 1bcg,2ab,7 (Optional: Use BU website -> Java Applets for Chaos and Fractals -> Linear Web to experiment and to check your answers to problem 7.)
5. Reproduce the table on p. 23 using a spreadsheet (such as EXCEL). All your numbers might not match exactly. Why? (Thought question - not to hand in.) If your printout is more than one page, provide only the first page.
6. Do the following tasks with the help of the BU Website by linking to "Java Applets for chaos and fractals" and then "Linear Web" or "Nonlinear Web" (from any computer with web access and JRE (Java Runtime Environment) installed:
  1. Experiment to see how to operate the Linear Web and Nonlinear Web software. (Nothing to hand in.)
  2. Do Experiment 3.6 on pages 25-6 of the text, including the essay and Notes and Questions 1 and 2 (but not 3). The book suggests using 10 initial conditions for each of the three functions, but you may use just 4. You may use either a spreadsheet or Nonlinear Web (or both). All the functions are available via a menu in Nonlinear Web. You can type the functions in your self in Excel. Hints: You can use the mod function to write the doubling function. Also for the doubling function, you might want to experiment with the number of decimal places printed. Make sure you iterate at least 70-100 times for each seed. You may use printouts to support your writeup, but you are not required to do so.
Set 2, Due Friday 9/30.
1. Read Devaney: Chapters 5-7.
2. Ch 5: 1bcj, 2abc, 3, 4ad, 5,7,9 (for f'(x_0)=+1 only; OK to assume the result of 8)
  Extra Credit: State and prove a neutral fixed point theorem analogous to problem 9, except assuming F'(x_0)=-1. (Big hint: Expand F in a power series around the fixed point, compute F(F(x)), and apply problem 9.)
3. Ch 6: 1aef (Suggestion: use software for part f), 3,4,5,6,7,8,9
4. Complete the following chart for periods n=1,2,...,10 for the quadratic map with c=-2 (Q_c(x)=x²+c) using the following column headings:
  n, # period-n pts, # prime period-n points, # prime period-n orbits.
5. Experiment 6.4 We will do this experiment together as a class. Each person should record the eventual behavior of four seeds on the class graph. Seeds for each person will be distributed in lab; a chart on which to plot your results will also be provided. See assigned parameter values here.
6. Do the lab tasks on tangent and period-doubling bifurcations. Handout in class or download here.
Set 3 Due WEDNESDAY 10/12. Extra Credit problems may be handed in through Friday, Oct. 21.
1. Read Devaney: Chapters 7-10.
2. Ch 7: 1,2,3,9,10,11,12,13 (OK to assume result of 9 to prove 12 and 13, so this is not a formal proof.) Extra Credit: 8.
3. Ch. 9: 1,2,5,7,8,9
4. Show that for any given real number r>0 there exists a c such that rx(1-x) is conjugate to y²+c. Hint: Try a conjugacy of the form h(x) = Ax + B. Solve for A,B,c in terms of r.
5. Lab Related Work:
  1. Ch. 8: Do Experiment 8.3: Windows in the Orbit Diagram. Do Notes and Questions 1,3,4. Extra credit: Question 5. Also extra credit: Ch. 8: 16,17.
  2. Attracting parameter intervals for the quadratic map. (This is an extension of the lab work from HW 2, # 6.) Use the Nonlinear Web/Orbit Diagram software at the BU web site and/or spread sheets and/or Mathematica to locate ALL parameter intervals of attracting prime-period-n orbits for n=1,2,3,4,5 for the quadratic map:Q_c(x)=x²+c. Restrict the parameter c to -2<c<1/4. Explain briefly how you obtained your answers.
    Hints:
    - Use the graph of the nth iterate to look for approximate parameter values where period-n orbits are born. The slider on the BU Nonlinear Web software is especially useful for changing parameter values. Or you can use the Manipulate command in Mathematica.
    - Use the approximate parameter values obtained in the previous step to magnify the appropriate region in the Orbit Diagram software. Locate the appropriate saddle-node and/or period-doubling bifurcations which correspond to the "birth/death/change of stability" of the attracting periodic orbit you are investigating. The disadvantage of this software is that you cannot read the mouse position off directly, and you cannot "back out" one step at a time from your magnifications. The Orbit Diagram software has the big advantage that you can choose a magnification region either with the mouse or by typing in window ranges. You are also allowed to change the number of transient iterates to ``hide'', and the number beyond the hidden iterates to ``display.''
    - You could find the endpoints of the attracting intervals by using only Nonlinear Web software, and doing graphical iteration to see whether orbits are drawn toward a specific periodic orbit or not. This, however, tends to be slow, and it is difficult to locate endpoints accurately.
Set 4 Due date Wednesday October 26. (Recommended - especially the reading - to be done before Test 1 on Friday Oct. 21.)
1. Read Devaney Ch. 10. Know the three properties for a dynamical system to be chaotic. Skim Ch's 11 and 12. Read carefully The Period 3 Theorem (p. 133), Sarkovskii's Theorem (p. 137, including the Sarkovskii ordering), and the negative Schwarzian derivative theorem (p. 158).
2. Extend your "Chart" from HW 2 to include the following. For n=1 to 10, fill in columns: n, # slns to Q_-2ⁿ(x)=x, # prime per-n pts for c=-2, # prime per-n orbits for c=-2, # of c-intervals corresponding to attr. per-n orbits, # of period-n orbits born (as c decreases) in period-doublings (from period n/2), # of period-n orbits born in saddle-node bifurcations, # of period-n windows.
3. Extra Credit: Ch. 10: 20 Prove directly only the property that periodic points are dense.
Test 1: Friday Oct 21. 8:20-9:50 in EduE 32. Chapters 1-9 and selected parts of Chapters 10 - 12 of Devaney. Select here for Midterm 1 topic list.
Set 5 Due Fri. Nov. 4.
1. Read Strogatz Ch's 1 - 3
2. Sec. 2.1: 1,2
3. Sec. 2.2: 3*,7*, 8, 9, 13c
  *For 3 and 7, determine equilibrium points (fixed points) (exact location for 3, approximate location for 7) and their stability (attracting, repelling or neither), sketch phase lines, and sketch corresponding solution curves.
4. Other sections in Ch. 2: 2.4.4, 2.4.7, 2.5.2, 2.6.1, 2.7.1, 2.8.2c (For 2.8.2c you may use software to draw the slope field, or sketch a slope field by hand. The Rutgers software allows 1D and 2D differential equations, plotting vector fields (slope fields for 1D) and printing. Mathematica, of course, allows everything, but you need to figure out how to use it. Try the VectorPlot command, for example. Sketch several solution curves by hand on the slope field.)
5. Ch. 3: 3.1.1, 3.2.1, 3.4.4. Extra Credit: 3.2.6.
6. Locate and identify all bifurcations in the family: dx/dt = x (r - (1 - x²)) (r - (2x³ - 2x)). It is OK to use software to plot any "useful" functions.
7. Change of variables:
  1. Logistic D.E.: dx/dt = ax-bx². Rescale by y=Ax. (A is a constant number.) Choose A so that the "new" differential equation (in y) is dy/dt = ay - ay².
  2. Logistic map: x_n+1=ax_n - bx_n². Rescale by y=Ax. Choose A so that the "new" map is y_n+1=ay_n - ay_n². (Subscripts corrected 11-4-11.)
Set 6 Due Mon. Nov. 21.
1. Read Strogatz Ch 5, sections 5.0, 5.1, 5.2
2. Read Strogatz Ch 6: 6.0-6.4, 6.72qq11
3. Read Strogatz Ch 7: 7.0, Example 7.1.1, 7.3 first paragraph (Poincare-Bendixon Theorem statement) and last paragraph (No Chaos in the Plane, p.210).
4. Read Strogatz Ch 8: 8.0, 8.1 up to beginning of Example 8.1.1 and the paragraph on Transcritical and Pitchfork bifurcations (p 246).
5. Read Strogatz Ch 9: (the Lorenz equations) 9.0, 9.2 (p. 311 only), 9.3 (p. 317-319 only) and Color Plate 2.
6. Read 12.2 (the Henon map)
7. 5.1: 3,5,7*,9a,c-f, 10ace; Extra Credit: 9b
  *5.1Notes:
  - Do 5.1 #7 by hand. Include both nullclines (dx/dt=0 and dy/dt=0) and any real eigenvectors in your sketch.
8. 5.2: 1,3*,6*,7*,10*, 11*, 13ab*; Extra Credit: 13c.
  *5.2 Notes:
  - In 5.2: 3,6,7,10: You may use software to obtain the phase portrait. Include several phase curves. Add arrows by hand to indicate the direction of travel as time increases. Include nullclines and any real eigenspaces in all phase portraits.
  - In 5.2 #11: Do not "solve" the system analytically. Do show the one-dimensional eigenspace and the phase portrait (including several direction vectors and several phase curves).
  - In 5.2 # 13: You may use software (Differential Systems) for your phase portraits. Sketch any equilibria, nullclines and real eigenspaces on the printout. If there are two eigenvalues with the same sign, indicate the "stronger" one with double arrows. (Or do it all by hand.) Hint: Think of the different regions of the "trace-determinant space"; you should have 2 main (not borderline) cases, depending on the parameters. Assume that the mass m>0 and the spring constant k>0.
9. In 6.1: 5, 8. For 5, you need not sketch the vector field; instead, include the direction field at least along the nullclines. For both 5 and 8, use the phase portrait to describe the fate of ALL orbits.
10. In 6.3: 16
11. In 6.4: 2
12. In 6.7: 1
13. In 8.1: 1a
14. In 8.2: 5
Set 7 Due Monday, Dec. 5 Select here for Midterm 1 topic list.
1. Devaney, Ch 15: 1d, 2e, 3e, 5ac, 6 (hint: translate), 8ac, 9, 11
2. Devaney, Ch 16: 1,3,5abc (for 5b, restrict c to be real),6a,9bc
Test 2, FRIDAY Dec 9, 8:20-9:50 in EduE 32. Covers Strogatz material and Devaney Chs 15, 16 and part of 17. Select here for Midterm 2 topic list. Note that this list expands the topics list for Midterm 1, but topics that are primarily for one-dimensional maps will not be asked again on Midterm 2. Select here for a practice test and here for solutions. The test is from 2007 before we used the Strogatz text, but the topics covered are still similar.
Set 8 Wed. Dec. 14
1. Devaney Ch 17: Lab Experiment 17.5 in Devaney. Indicate the location of the eight c values on the copy of the Mandelbrot set which was handed out in lab. Answer Notes and Questions #2. #3 may be done for extra credit.

Math 5260 Final Problem Set Due Wednesday, Dec. 21 at noon. Do problems 1-7. 100 points total.
Ground rules for the final problem set. Unilike for HW sets during the semester, you are not allowed to collaborate for the final problem set. You may use software, computer programs, books, internet, but any use of these references must be fully documented!!!!! You may ask me questions, but no other person.

Do All problems 1-7. Problems 5-7 have choices. 100 points total.

(15pts) Analyze the dynamics and bifurcations of the family:

dx/dt = ax - xy
dy/dt = -y + x^2.
(30pts) Analyze the dynamics and bifurcations of the family:

x_n+1=a(x_n-x_n³/3). Both x and c are real.

Hints:
1. Note that Boston University software includes this family.
2. In addition to the other standard behaviors you might look for, determine which orbits stay bounded, which bounded orbits stay positive, and which bounded orbits stay negative. Determining various invariant - or noninvariant - intervals should also help.
3. Remember the goal of analyzing a family of dynamical systems is to describe the behavior all orbits, and how that description changes as parameter(s) are varied. Part of your description can be to compare/contrast the dynamics of this family to that of a "known" family, such as x^2+c.
(15pts) Number of period-n attracting intervals and decorations. (Use of software like Mathematica is encouraged.)
1. Determine the number of (prime) superattracting period-n orbits in the family x->x²+c for n=1,2,3,4,5,6. For n=1,2,3,4,5, locate and label the superattracting period-n point having x coordinate zero on the orbit diagram printout. Explain how you obtained your answers.
2. Determine the number of superattracting period-n orbits in the family z->z²+c, for n=1,2,3,4,5,6. For n=1,2,3,4,5, locate and label the parameter value of each superattracting period-n orbit on the Mandelbrot set printout. Explain how you obtained your answers.
(20pts) Fixed, prefixed, and preperiod-2 points in x->x^2+c and z->z^2 + C. Justify all answers. The Boston University Mandelbrot/Julia Set software will be useful to at least graphically check your answers. If you cannot compute exact answers, the software can be used via trial and error to find sought-after points.

(A) Consider the map Q(x) = x^2-1. (x is real.)
(i) Compute the two fixed points p₊ and p_-. Both should be repelling. Verify this. Show that there is also a superattracting orbit between x=0 and x=-1.
(ii) Compute all points (excluding p₊) that are eventually fixed at p₊. Hint: there is only one! You may compute it exactly or compute a decimal approximation.
(iii) Using graphical iteration, locate any 5 points that are eventually fixed at p_-. You need not compute their values. Choose them so they are not ALL on the same orbit. Hint: there is an infinity of them!
(iv) Using graphical iteration, locate any 3 points (excluding x=0 and x=-1) that are eventually period-two. Recall that eventually period-two points must land exactly ON either x=0 or x=-1, not just be attracted toward the period-two orbit.
(v) Label all nine of your preperiodic points from parts (ii), (iii), and (iv) as a₁, b₁, ..., b₅, c₁, c₂, c₃, respectively, on a graphical iteration diagram.

(B) Consider the map Q(z) = z^2-1. (z is complex.)
(i) Compute the two fixed points p₊ and p_-. Both should be repelling. Verify this. Show that there is also a superattracting orbit between z=0 and z=-1.
(ii) Compute any one NONREAL point that is eventually fixed at p₊. Hint: there is an infinity of them in general, but no NONREAL points which land on p₊ in fewer than two iterates! You may compute it exactly or compute a decimal approximation, or use the Mandelbrot/Julia Set Applet to determine its location on the picture of the z-plane.
(iii) Compute any one NONREAL point that is eventually fixed at p_-. A decimal approximation is adequate. Hint: there is an infinity of them, but no NONREAL points which land on p_- in fewer than three iterates.
(iv) Compute any one NONREAL point that is eventually period-two.
(v) Label all nine of your REAL preperiodic points from parts A(ii), A(iii), and A(iv) as a₁, b₁, ..., b₅, and c₁, c₂, c₃, respectively, on the Julia Set diagram provided for z^2-1. Label the three preperiodic points from parts B(ii), B(iii), and B(iv) as A, B, and C, respectively, on the Julia Set diagram provided in class.
(vi) Describe briefly the fate of all initial conditions for the map z^2-1.

C. (Extra Credit: 5pts) Find analagous preperiodic points on the Julia set for Douady's "rabbit" on the diagram provided. You may compute them or use iteration software to locate them. In either case, locate them on the graph of the rabbit Julia set.
(6 pts) Attend class Mon. Dec. 12 (Test 2 review, Hints for Final Problem Set, and Lab 17.5) OR
For the 3D Lorenz system, fix sigma at 10, b at 8/3. Let r vary. Construct a bifurcation diagram in x and r indicating the values of equilibria for r in (0,30). Compute the stability of the equilibria: sink saddle or source. Indicate stable by solid lines, saddle or source by dashed lines. Are any equilibria attracting at the classic parameter value of r=28?
(7 pts) Attend class Wed. Dec. 14 (Newton's method and related projects) OR
Derive the formula for Newton's method for the complex function F: N(z)=z-F(z)/F'(z), where z is the guess for a root of F, and N(z) is the new "improved" guess by taking the linear approximation (tangent line if real) at (z, F(z)). Show that zeroes of F are superattracting fixed points of N. Show N for F(z)=z²+1 is conjugate to Q(z)=z². (See problem 2, Chapter 18.5 in Devaney for more details.)
(7 pts) Attend class Fri. Dec. 16 (Henon attractor related projects) OR
Consider the following version of the Henon map: (x,y)->(a-by-x^2, x). Show that the map is invertible (compute the inverse). Find all fixed points and linearize around the fixed points. What are the eigenvalues of the linearizations?

Class Schedule. This information will be updated by the instructor throughout the course.

Week	Monday	Wednesday	Friday
1 Sept. 5-9	Labor Day No School	Intro	Intro cont. Devaney ch 1-4
2 Sept. 12-16	Ch 4 HW 1 Lab (bring laptops to class) including Expt. 3.6	Dev Ch 5	Dev Ch 5 HW 1 due
3 Sept. 19-23	Dev Ch 5	Dev Ch 6-7	Dev Ch 6-7 HW 2 lab work Bring laptops to class
4 Sept. 26-30	Chs 6-7 continued	Dev Ch 7	Dev Ch 8 HW 2 due
5 Oct. 3-Oct.7	Ch 8 HW 3 Lab	Ch 9	Long class:8:35-9:50 Ch 9
6 Oct. 10-14	Dev Ch 10	Dev Ch 10 HW 3 due	NO CLASS
7 Oct. 17-21	Dev Ch 10	Review	Test 1 8:20-9:50
8 Oct. 24-28	Test comments Strogatz Ch 1-2	Stogatz Ch 2 HW 4 Due	Long class: 8:35-9:50 Strogatz Ch 2
9 Oct. 31-Nov.4	Strogatz Ch 3	Strogatz Ch 3	Strogatz Ch 3 HW 5 Due
10 Nov. 7-11	Strogatz Ch 5	Strogatz Ch 5	Strogatz Ch 5
11 Nov. 14-18	Strogatz Ch 6	Strogatz Ch 6	Strogatz Ch 6
12 Nov. 21-25	Strogatz Ch 7 HW 6 due	Strogatz Ch 8	Thanksgiving
13 Nov. 28-Dec.2	Devaney Ch 15	Devaney Ch 15-16	Devaney Ch 16-17
14 Dec. 5-9	Devaney Ch 17 HW 7 Due	Review for Test 2	Test 2 8:20-9:50
15 Dec. 12-16	Lab Expt 17.5 Bring laptops to class	Course summary/ Further research topics HW 8 due	Course summary/ Further research topics
16 Dec. 19-23		Final Project/HW Set Due Wed. Dec. 21, noon

This page (http://www.d.umn.edu/~bpeckham/www) is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Thursday, 08-Dec-2011 19:05:23 CST.