Bruce B. Peckham - Fall 2009 Dynamical Systems (Math 5260)

Math 5260 Dynamical Systems

Fall 2009 semester
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. :) )
Hirsch, Smale, Devaney text homepage, with link to errata
Check grades recorded to date.

Homework Assignments

Set 1 Due Wednesday 9/16/2009: (DUE DATE MOVED to FRI 9/18/09)
1. Read Hirsch, Smale and Devaney: Preface and Chapter 1 through Section 1.2.
2. Read Devaney: 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 with 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 suggest 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 Wednesday 9/30/2009 Due date moved to Friday 10/2/2009.
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 three 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.
6. Do the lab tasks on tangent and period-doubling bifurcations. Handout in class or download here.
Set 3 Due Friday 10/16/2009. Extra Credit problems may be handed in through Monday, Oct. 26.
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 Monday October 26. (Recommended - especially the reading - to be done before Test 1 on Thursday October 22.)
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: Thursday Oct. 22. 6:30-8:00. Room Bohannon Hall 90. Chapters 1-9 and selected parts of 10 - 12 of Devaney. Select here for Midterm 1 topic list.
Set 5 Due Thursday, Nov. 5. (No class Friday Nov. 6)Due date moved to Wednesday, Nov. 11. Extra credit may be turned in any time until Monday, Nov. 16.
1. Read Hirsch-Smale-Devaney (HSD)
  - Ch 1: Read secs 1.1,1.2,1.3; skim Sec. 1.4, especially for the definition of a Poincare map.
  - Ch 2: Intro (pp 21-22), 2.1, 2.2, skim 2.3, 2.4, review 2.5, skim 2.6, skim 2.7.
  - Ch 3: Sections 3.1-3.3, skim 3.4
  - Ch 14: Intro (pp 303-4) and sec. 14.1. Skim 14.2-14.5.
2. Ch 1 Exercises: 2a, 3 (include one phase line for each distinct parameter region), 4, 13
3. Ch 2 Exercises: 1ac, 3, 7
4. Ch 3 Exercises: 1
5. Change of variables: NOTE: A is a real constant, NOT a matrix, in this problem.
  1. Logistic D.E.: dx/dt = ax-bx². Rescale by y=Ax. 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.
6. Related to chapter 14: Sketch the beginnings of a bifurcation diagram for the Lorenz equations. Keep track only of equilibrium points. Fix sigma=10 and b=8/3. Plot the x coordinate of the equilibria vs r.
7. Extra Credit. Exploration 1.6. Do any or all of questions 1-6.
Set 6 Due Friday, Nov. 20
1. Read Hirsch-Smale-Devaney (HSD)
  - Ch 4: Determinant-Trace space and classification of equilibria. Secs 4.1; skim Secs 4.2 - espcially for defs of conjugate and hyperbolic.
  - Ch 5: Linear Algebra. Skim Sec. 5.2. I will not expect that you know material in this Chapter.
  - Ch 6: Higher dimensional linear systems (especially 3D). None of this is required, but you might want to look at the four examples in Sec. 6.1 corresponding to Figures 6.1 - 6.5.
  - Ch 7: Nonlinear Theory. Skim. Not required to know.
  - Ch 8: Linearization near equilibria. Read Secs 8.1, 8.4, 8.5. Skim 8.2, especially for Linearization Theorem. Skim 8.3, especially for the Stable Curve Theorem.
  - Ch 9: Read 9.1 - Nullclines.
  - For extra credit problems only: Ch 14: Reread Intro (pp 303-4) and sec. 14.1. Read 14.2 and 14.3. Skim 14.4-14.5.
2. Ch 4 Exercises: 1, 3 (For each distinct "region" in the k-b parameter space, interpret your phase portrait in terms of a spring-mass system.)
3. Ch 8 Exercises: 5
4. Change of variables. Consider the system of differential equations defined by x'=ax-by, y'=bx+ay. Change to polar coordinates, using r²=x²+y², and tan(theta)=y/x. Show that the resulting differential equation system is r'=ar, and theta'=b.
5. Extra Credit: Ch 14 Excercises: 1
6. Consider the following system of differential equations representing two competing species: x' = x(2-x-y), y'=y(3-2x-y)
  1. Find all equilibria
  2. Linearize around each equilibrium. Use the matrix of linearization to compute the eigenvalues of each equilibrium and classify each equilibrium as sink, saddle or source.
  3. Include a graph of the nullclines superimposed on your phase plane. These are curves in the phase plane corresponding to x'=0 OR y'=0. Note that equilibria occur where these curves intersect. Determine the direction field along all nullclines. (You can check your calculations using the Penn State software, or you can check it yourself with Mathematica.)
  4. Extra Credit. Compute the eigenvectors of each linearization which has real eigenvalues. You can do this with Mathematica or other software if you wish. The eigenvectors for any saddles are particularly useful. Use the linearizations to help determine the phase portrait of the nonlinear system near each equilibrium point.
  5. Sketch a phase plane for this system. Use a combination of information from your linearizations, and nullclines. You may check your sketches using software. Label clearly all equilibria. Identify and label the stable and unstable manifolds of any saddles.
  6. Interpret the long term behavior of the system in terms of the two populations x and y. You can restrict your attention to the first quadrant.
Set 7 Due Friday, Dec. 4.
1. Read Devaney Chapters 15, 16, 17.
2. Devaney, Ch 15: 1d, 2e, 3e, 5ac, 6 (hint: translate), 8ac, 9, 11
3. Devaney, Ch 16: 1,3,5abc,6a,9bc
4. 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 will be handed out in class. Answer Notes and Questions #2. #3 may be done for extra credit.
5. More TBA.
Test 2 Wednesday Dec. 9, 6:30 - 8pm in TBA. Select here for Midterm 2 topic list.
Final Problem Set Due Monday, Dec. 21.
____________________________________
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This page is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Monday, 23-Nov-2009 17:54:21 CST.