Bruce B. Peckham - Fall 2007 Dynamical Systems (Math 5260)

Math 5260 Dynamical Systems

Fall 2007 semester
Prof. Bruce Peckham, Department of Mathematics and Statistics, University of Minnesota, Duluth

Homework Assignments

Set 1 Due Friday 9/14/2007:
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).
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.
  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).
Set 2 Due Friday 9/28/2007
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.
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 in Lab. 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 lab on Friday Sept. 21 or download here.
Set 3 Due Friday 10/12/2005
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: 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. Locate the appropriate saddle-node and/or period-doubling bifurcations which correspond to the "birth/death/change of stability" of the attracting periodic orbit.
    - Optional. Corroborate your bifurcation values with graphical iteration. The BU Nonlinear web may work for some of the bifurcation values, but since you can't type in an arbitrary parameter, you won't be able to check all the bifurcations.
    - Extra Credit. Look for the logistic map as a Mathematica Demonstration contribution in Mathematica 6. Download it and change the iteration function from the logistic map to the quadratic map. Then use graphical iteration to corroborate the attracting intervals you located using the orbit diagram.
Set 4 Due date Wednesday October 24.
1. Read Devaney Ch. 10. 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. Ch. 10: 20 for periodic points dense, Extra Credit: also show SIC and Transitivity.
Test 1: Friday Oct. 26. 8:30-9:50. Chapters 1-9 and selected parts of 10 - 12 of Devaney. Select here for Midterm 1 topic list.

Optional: Test corrections. Due Monday Nov. 5.
Redo full problems on a separate sheet. Hand in with original test. 1 point granted for every 3 points of corrections made.
Set 5 Due Wednesday, Nov. 7
1. Read Hirsch-Smale-Devaney (HSD)
  - Ch 1: 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, 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 and their stability (solid lines for stable, dashed lines for unstable). Vary only parameter r. Fix sigma=10 and b=8/3. Plot the x coordinate of the equilibria vs r. UPDATE: Forget the solid/dashed distinction for stability for now. It will be assigned later.
7. Extra credit. Locate and identify all bifurcations in the family: dx/dt = x (r - (1 - x²)) (r - (2x³ - 2x)). You may want to use Mathematica to plot all the equilibrium solutions.
Set 6 Due Wednesday, Nov. 21
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). Read 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 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. Ch 6 Exercises: 3
4. Ch 8 Exercises: 5
5. 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. Sketch a phase plane for this system. Use a combination of information from your linearizations, and software. Label the equilibria. Identify and label the stable and unstable manifolds of any saddles.
  4. 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.
  5. 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. (The Penn State software does this for you.)
  6. Extra Credit. Compute the eigenvectors of each linearization which has real eigenvalues.
Set 7 Due Wednesday, Dec. 5. Corrections Due Wed. Dec. 12. Redo full part of problem even if you aleady received partial credit. Attach original graded problem set.
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 Extra Credit (Due Wed. Dec. 12) 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. Extra Credit from the geometric Lorenz attractor. Figure 14.8 shows the image of the "sigma" plane as two triangular regions. Superimpose on the original plane and the two triangular regions a sketch of the image of the two triangular regions under the Poincare map.
Test 2 Friday Dec. 7, 8:30 - 9:50am in LSci 170. RESCHEDULED: Tuesday Dec. 18, 12-1:50. Usual classroom. Select here for Midterm 2 topic list.
Final Problem Set Due Wednesday, Dec. 19.
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SPECIAL GROUND RULES: For this final problem set, do not work with each other. Do not ask anyone besides me for help.
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Do problems 1-4 and either 5 or 6. 60 points total.
1. (10pts) Analyze the dynamics and bifurcations of the family:
  
  dx/dt = ax - xy
  dy/dt = -y + x^2.
2. (20pts) Analyze the dynamics and bifurcations of the family:
  
  x_n+1=a(x_n-x_n^3/3).
  
  Rather than describing all dynamics and bifurcations from scratch, concentrate on similarities and differences from the bifurcations of the quadratic family. You may assume that the dynamics and bifurcations of the quadratic family is "known."
  Hint: In addition to the other standard behaviors you might look for, determine which orbits stay bounded, which bounded orbits stay positive, and which stay negative. Determining various invariant - or noninvariant - intervals might also help. Note that Boston University software includes this family.
3. (10pts) Number of period-n attracting intervals and decorations. (Use of software like Mathematica is encouraged.)
  1. Determine the number of 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 diagram. Explain how you obtained your answers.
4. (5pts) Do one of the following, but not both:
  1. Devaney, Ch 17 Ex. 3: show that the Mandelbrot set is symmetric about the real axis.
  2. Lorenz equations. For the 3D Lorenz equations with the "standard parameter values" (sigma=10, b=8/3, r=28), locate all equilibrium points and determine their stability, including the number of stable/unstable eigenvalues at each equilibrium.
5. (15pts) Attend class on MWF the last week of classes.
6. (15pts) Devaney, Chapter 11, problem 6: show the given map has an orbit of period 7 but no orbit of period 5; AND problem 7: show the given map has periodic orbits of all even periods, but no periodic orbits of any odd period.
7. Make a deal. Do other problems from either text, write computer programs to do experiments, or pursue further something related to the course that especially interested you. Clear the substitutions with me first.
- Go to Bruce Peckham's Home Page
- Go to the Department of Mathematics and Statistics Home Page
This page is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Tuesday, 08-Sep-2009 13:22:10 CDT.