Ordinary Differential Equations and Stability Theory
Fall 2006

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MATH 454 - Fall 2006 - Assignments

D2L Course Page

• Due Tuesday, August 29th:
  1. Read Chapters 1 and 2 of the book.
  2. Search the recent scientific literature for problems modeled by dynamical systems, and describe two different examples. For each of these examples,
     • Explain what the problem is;
     • Define the variables (x₁,x₂,...) and say what they represent;
     • Write down the dynamical system and define any relevant parameters;
     • Answer the following questions: is the dynamical system autonomous? Is it linear or non-linear? What is its dimension? In each case, justify your answer.
     • Do not forget to quote the book or scientific paper from which you obtained this information.
     Your answers should be typed and put in my D2L drop box.



• Quiz on Tuesday, September 5th: Problems # 2.1.1, 2.1.2, 2.1.3 p. 36; # 2.2.3, 2.2.8, 2.2.9, 2.2.10 p. 37; # 2.3.2, 2.3.4 p. 39; # 2.4.2, 2.4.5, 2.4.7, 2.5.3 p. 40; # 2.6.1, 2.6.2 p. 41; # 2.7.6, 2.7.7 p. 42.



• Quiz on Tuesday, September 12th: Problems # 3.1.1, 3.1.3 p. 79; # 3.2.2, 3.2.4, 3.2.5 p. 80; # 3.4.2, 3.4.4 p. 82; # 3.4.6, 3.4.7, 3.4.9, 3.4.14 p. 83; # 3.4.16 p. 84.



• Due Monday, September 18th: Please fill out the D2L questionnaire on the level of difficulty of homework problems. You can find this questionnaire in the "Quizzes" section of the D2L course page. It is called Chapters 2-3. You will not be able to access this questionnaire after 12 noon on Monday.



• Homework due on Thursday, September 21st (no in-class quiz; I will collect the homework and grade one or two problems): Problems # 3.5.8, 3.6.2, 3.6.3, 3.6.4 p. 86; # 3.6.7 p. 88; # 3.7.4 (b-f) p. 90.



• Quiz on Tuesday, September 26th: All of the problems in the Linear Algebra Problem Set.



• Quiz on Tuesday, October 3rd: All of the problems in the Linear Algebra / ODE Problem Set.



• Test on Tuesday, October 10th: Review all of the material discussed so far.



• Due on Thursday, October 12th: The work you did in the computer lab. Please type or scan your answers, include figures obtained with PPLANE and/or Plotter, and put your papers in my D2L drop box by midnight on the 13th. Alternatively, you can bring a paper copy of your work to the Math Main Office (Room 108) and ask that it be put in my mailbox.



• Quiz on Tuesday, October 17th: Problems # 5.1.2, 5.1.4, 5.1.5, 5.1.9, 5.1.10 p. 141; # 5.2.4, 5.2.7, 5.2.9, 5.2.10, 5.2.11, 5.2.13 p. 143.



• Quiz on Tuesday, October 23rd: Problems # 6.1.3, 6.1.4, 6.1.10, 6.1.11 p. 181; # 6.2.1, 6.2.2, 6.3.1, 6.3.2, 6.3.4, 6.3.7 (i.e. use PPLANE to check your answers to the preceding problems) p. 182; # 6.3.11 p. 183.



• Quiz on Tuesday, October 31st: Problems # 6.5.4, 6.5.5, 6.5.6 p. 186; # 6.5.9 p. 187; # 6.5.13 p. 188; # 6.5.19 p. 189; # 6.6.1, 6.6.3, 6.6.4, 6.6.5 p. 190. For a discussion of how to plot the phase portrait of a conservative system with potential V(X), see the notes on the nonlinear pendulum, posted on the D2L course page.



• Quiz on Tuesday, November 7th: Problems # 6.6.6 p. 190; # 6.7.2 p. 192; # 6.8.3, 6.8.7 p. 193; # 6.8.13, 6.8.14 p. 194.



• Quiz on Tuesday, November 14th: Problems # 7.1.1, 7.1.5, 7.1.8 p. 228; # 7.3.1, 7.3.2, 7.3.4 p. 231.



• Quiz on Tuesday, November 21st: Problems # 7.2.5, 7.2.6 p. 229; # 7.2.10, 7.2.12, p. 230; # 7.2.15 p. 231; # 7.6.2 p. 235; # 7.6.19, 7.6.20 p. 238.



• Quiz on Tuesday, November 28th: Problems # 8.2.5, 8.2.6, 8.2.7, 8.2.8 p. 287; # 8.2.12, 8.2.16 p. 289.



• Homework due Tuesday, December 5th (no in-class quiz. I will collect and grade the homework).
  Your papers will be accepted until 4 pm on Thursday, December 7th (Dead Day). Please bring them to the Math Office (Room 108).
  1. Make near-identity changes of variables in the system of problem 8.2.5 to derive the normal form for the corresponding Hopf bifurcation. Use this information to calculate the amplitude of the limit cycle as a function of the control parameter. Check your answer with PPLANE.
     You may consult the following lecture notes on the derivation of the normal form of a Hopf bifurcation.
     Solution.
  2. Re-write the system of problem 8.2.8 in a form suitable for the analysis of the Hopf bifurcation that the fixed point with coordinates (a,a-a²) undergoes at a = a_c = 1/2. More precisely, transform the system so that it is in a form appropriate for applying the analytic criterion of problem 8.2.12.
     Hint: in problem 8.2.12, the fixed point of interest is the origin, the system is written at the bifurcation point, and its Jacobian is anti-diagonal.
     Solution.
  3. Write the system of problem 8.2.7 in terms of a single variable z = x + i y. Make a near-identity change of variable to eliminate the term in |z|², and keep track of the cubic terms that this transformation generates.
     Hint: remember that the unknown coefficient in the near-identity change of variable is a priori complex.
     Solution.
  
  
  
• Final exam on Tuesday, December 12th: Review all of the material covered during the semester.

A compilation of all of the quizzes of the semester