Math 5260 Dynamical Systems

Fall 2017 Course Homepage
Prof. Bruce Peckham, Department of Mathematics and Statistics, University of Minnesota, Duluth

Homework Assignments

Set 1, due FRI Sept. 1

  1. Strogatz: Read Preface and Chapter 1: Overview.
  2. Devaney: Read Preface and Ch's 1-4
  3. Devaney Ch 3: 1,3,4,5,6,7abh,8,11,12,13,14
  4. 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.
  5. 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 5. 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 yourself 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 Sept 8
  1. Devaney 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.)

Set 3, Due Friday Sept 15.
  1. Read Devaney: Chapters 5-6.
  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. Complete the following chart for periods n=1,2,...,10 for the quadratic map with c=-2 (Qc(x)=x2+c) using the following column headings:
    n, # period-n pts, # least period-n points, # least period-n orbits. Hint: You can use Nonlinear web for n<=6, but you will need to think of the properties of the graph of the nth iterate for higher iterates.

Set 4, Due Friday Sept 22.
  1. Read Devaney: Chapters 6-7.
  2. Ch 6: 1aef (Suggestion: use software for part f), 3,4,5,6,7,8,9 (Sketch the bifurcation diagram only; plot attracting fixed points with solid lines, repelling fixed points with dashed lines.)
  3. 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 class; a chart on which to plot your results will also be provided. Assigned parameter values are provided here.
  4. Do the lab tasks on tangent and period-doubling bifurcations. Handout in class or download here.

Set 5 Due Fri. Sept. 29.
  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. 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.
  4. 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 Set 3, # 3.) 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 least-period-n orbits for n=1,2,3,4,5 for the quadratic map:Qc(x)=x2+c. Restrict the parameter c to -2<c<1/4. Explain briefly how you obtained your answers.

Set 6 Due Wed. Oct. 4
  1. Reread Ch 9.
  2. Ch. 9: 1,2,5,7,8,9

Test 1: Monday Oct 9. 5-6:30 Room: SCC 120. Chapters 1-9 and selected parts of Chapters 10 - 12 of Devaney. Select here Midterm 1 topic list. See a practice test here.

Set 7 Due Fri. Oct. 13
  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 "Charts" from HWs 3, 4 and 5 to include the following. For n=1 to 10, fill in columns: n, # slns to Q-2n(x)=x, # least per-n pts for c=-2, # least per-n orbits for c=-2, # of c-intervals corresponding to attr. least per-n orbits (as c decreases), # of least period-n orbits born (as c decreases) in period-doublings (from period n/2), # of least period-n orbits born in saddle-node bifurcations, # of least period-n windows.
  3. Extra Credit: Ch. 10: 20 Prove directly only the property that periodic points are dense for the doubling function.

dSet 8 Due Fri Oct. 20.

  1. Read Strogatz Ch's1 - 2
  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 print the slope field, or sketch a slope field by hand. The Rice University software plots slope fields for 1D differential equations. It also allows selecting initial conditions and plotting the corresponding solution curve, but I want you to do this part by hand. 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. Check with StreamPlot command.)

Set 9 Due Wed. Oct. 25.
  1. Read Strogatz Ch 3
  2. Ch. 3: 3.1.1, 3.2.1, 3.4.3. In all three problems, replace "sketch vector fields" with "sketch phase lines".
  3. 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.
  4. 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: xn+1=axn - bxn2. Rescale by y=Ax. Choose A so that the "new" map is yn+1=ayn - ayn2.


Set 10 Due Fri. Nov. 3
  1. Read Strogatz Ch 5, sections 5.0, 5.1, 5.2
  2. 5.1: 3,5,7*; EXTRA CREDIT 10ace, Extra Credit: 9
  3. 5.2: 1,3*,6*,7*,10*, 13ab*; EXTRA CREDIT: 11*, Extra Credit: 13c.
    *5.2 Notes:

Set 11 Due Fri. Nov. 10
  1. Read Strogatz Ch 6: 6.0-6.4, 6.7
  2. Read Strogatz 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).
  3. 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).
  4. Read Strogatz Ch 9: (the Lorenz equations) 9.0, 9.2 (p. 311 only), 9.3 (p. 317-319 only) and Color Plate 2.
  5. Read 12.2 (the Henon map)
  6. 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.
  7. In 6.3: EXTRA CREDIT 16
  8. In 6.4: 2
  10. In 8.1: 2 Change the directions to: Plot a bifurcation diagram showing the x-value of the equilibria as a function of the parameter mu. Make the curve solid where the equilibrium is attracting, and dashed if it is unstable (either a saddle or repelling). Sketch three representative phase portraits, one each for mu>0, mu=0, and mu<0. Describe the fate of all orbits for all three cases. What type of bifurcation is this?
  11. In 8.2: 5 Use software (Rice University) to plot three phase portraits: mu>0, mu=0, mu<0. Describe the fate of all orbits in all three cases. >
  12. For any phase plane in this assignment, include computation of linearizations around all equilibrium points to classify as sink/saddle/source, with sinks and sources subclassified as nodes or spirals, stable/unstable manifolds of all saddle points, andnullclines.

Set 12Due Fri. Nov. 17
  1. Read Devaney Chapters 16 and 17
  2. Devaney, Ch 15: 1d, 2e, 3e, 5ac, 6 (hint: translate), 8ac, 9, 11
  3. Devaney, Ch 16: 1,3,5abc (for 5b, restrict c to be real),6a,9bc

Test 2, Tues. Nov. 21, 5-6:30 Room TBA. 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 an older practice test with slightly different topics, but with solutions. The second practice test is from 2007 before we used the Strogatz text, but the topics covered are still similar.

Set 13 Due Friday Dec 1.
  1. Read Devaney Ch 17.
  2. Do 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.

Final Problem Set Due Thursday, Dec. 14 at 3pm. Problems TBA.
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