MA522 Qualitative Theory of Differential Equations Page;

Fall 2009; Class 18014

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Announcements

• Solution to 3B in the practice.
• Practice for the second midterm.
• Solutions to the last quiz.
• My Syllabus for this class.

General Course Information

• Lecturer: Prof. Alexey Kuznetsov
• Office: LD 224 T
• Phone: (317) 278-7460
• Course syllabus, which are also available in the PDF format
• Lecture Times: Monday, Wednesday 4:30-5:45 PM in BS 3011
• Office Hours: Monday, Wednesday 3:30 PM -- 4:30 PM.

Homework Assignments/Preliminary Program

• Lecture # 1 on 8/26:
• In class: Definitions of a dynamical system, phase space, phase point and a trajectory. Considering dynamical system as a flow or vector field. One-dimensional dynamical systems: a qualitative geometric way of solving, the definition of equilibrium states and their stability.
• Read sections 2.0, 2.1, and couple of pages 2.2 (including ex. 2.2.1).
• Do exercises 2.1.1-3, 2.1.5, 2.2.1, 2.2.3, 2.2.5, 2.2.7
• Lecture # 2 on 8/31:
  • In class: Linearization, characteristic time-scale. Existence and uniqueness of solutions.
  • Read sections 2.4 and 2.5
  • Do exercises 2.4:1,3,5,7;   2.5:4 and 5
• Lecture # 3 on 9/2:
  • In class: Impossibility of oscillations in one dimensional systems. Potentials. Definitions of a bifurcation and a bifurcation point. The saddle-node bifurcation of equilibrium states in one-dimensional systems. Bifurcation diagrams. The notion of a normal form.
  • Read sections 2.6, 2.7, 3.0, and 3.1.
  • Do Exercises 2.6:1 and 2;   2.7:1 and 3;   3.1:1,3,5.
• Lecture # 4 on 9/9:
  • In class: Robustness of a dynamical system. Transcritical and pitchfork bifurcations of equilibrium states.
  • Read sections 3.2 and 3.4.
  • Do Exercises 3.2:1 and 3;   3.4:1,3,5,7,14,16.
• Lecture # 5 on 9/14:
  • Imperfect bifurcations and catastrophies.
  • Read 3.6
  • Do 3.6: 1,2,3,4.
• Lecture # 6 on 9/16:
  • Vector field on the circle.
  • Read 4.0-4.3
  • Do 4.1: 1,2,3,5;   4.3: 3,5.
• Lecture # 7 on 9/23:
  • Two-dimensional flows: major definitions.
  • Read 5.0 and 5.1
  • Do 5.1: 1,2,3,5,9,10(a,c,e),13.
• Lecture # 8 on 9/28:
  • Classification of equilibrium states for two-dimensional flows.
  • Read 5.2.
  • Do 5.2: 3,5,7,9.
• Lecture # 9 on 10/5:
  • Boundary types of equilibrium states. Phase portraits for nonlinear 2-D systems.
  • Read 6.0 and 6.1.
  • Do 5.2: 11,13;   6.1: 1,3,5.
• Lecture # 10 on 10/7:
  • 2-D phase portraits continued. Existence and uniqueness of solutions. Linearization of 2-D systems.
  • Read 6.2 and 6.3, but skip example 6.3.2.
  • Do 6.1: 7,9,12,13;   6.2: 1;   6.3: 1,3,5,9,16.
• Lecture # 11 on 10/12:
  • Sketching phase portraits of nonlinear 2-D systems.
  • Presentations: Predator-prey model.
  • Read 6.4.
  • Do 6.4: 1,3.
• Lecture # 12 on 10/14:
  • Practice for the first midterm exam.
  • Review all homework assignments and quizzes for the exam.
• Lecture # 13 on 10/19:
  • Midterm Examination
• Lecture # 14 on 10/26:
  • Conservative systems.
  • Read 6.5
  • Do 6.5: 1,2,3,6,8,12;   
• Lecture # 15 on 10/28:
  • Reversible systems.
  • Read 6.6
  • Do 6.6: 1,2,3,6,7,10.
• Lecture # 16 on 11/2:
  • Pendulum. Index theory.
  • Read 6.7 and 6.8
  • Do 6.7: 1(b>0, b<0),2;   6.8: 1.
• Lecture # 17 on 11/4:
  • Index theory.
  • Read 6.8
  • Do 6.8: 2,6,7,8,9,10.
• Lecture # 18 on 11/9:
  • Limit cycle.
  • Read 7.0 and 7.1
  • Do 7.1: 1,3,6,7,8.
• Lecture # 18 on 11/11:
  • Ruling out closed orbits: Gradient systems and Liapunov functions
  • Read 7.2
  • Do 7.2: 1,4,5,6,9,10,11,12.
• Lecture # 19 on 11/16:
  • Ruling out closed orbits: Dulac's criterion. Poincare-Bendixson Theorem; Glycolytic oscillator
  • Finish 7.2 and Read 7.3 and 7.4
  • Do 7.2: 13.
  • Do 7.3: 1,3,5,6,7
  • Do 7.4: 1,2.
• Lecture # 20 on 11/18:
  • Practice for the first midterm exam.
  • Review all homework assignments and quizzes for the exam.
• Lecture # 23 on 11/23:
  • Second midterm exam

Quizzes

• Quiz # 1 on Mon. 8/31
• Covers 2.0-2.2
• Solutions.
• Quiz # 2 on Mon. 9/14
  • Covers 2.4-2.6, 3.0-3.2, 3.4.
  • Solutions
• Quiz # 3 on Mon. 9/21
  • Covers 3.6, 4.1, and 4.3.
  • Solutions
• Quiz # 4 on Mon. 10/5
  • Covers 5.2.
  • Solutions
• Quiz # 5 on Mon. 10/5
  • Covers 6.1 and 6.3.
  • Solutions
• No Quiz in Mon. 10/26
• Quiz # 6 on Mon. 11/2
  • Covers 6.5 and 6.6.
  • Solutions
• Quiz # 7 on Mon. 11/9
  • Covers 6.7 and 6.8.
  • Solutions
• Quiz # 8 on Mon. 11/16
  • Covers 6.8, 7.1, and 7.2.
  • Solutions

Exams

• The Midterm Exam #1 is scheduled on October 19.
  • Practice
  • Solutions
• The Midterm Exam #2 is scheduled on November 23.
  • Practice

The dates for the midterms may change, and, if this occurs, I will announce the new dates as soon as possible.

• The Final Exam is scheduled for December 21, from 3:30 PM to 5:30 PM. The place will be assigned later.

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