Course Profile for MATH4402 Advanced Ordinary Differential Equations

(2 unit, 3L 1T)

Course Objective

• To build on the knowledge from MP213 (Ordinary Differential Equations) with greater emphasis on theoretical aspects of the subject. After initial discussion of existence, extendability and differentiabililty with respect to parameters of solutions we discuss stability and existence of periodic solutions. Students should gain an appreciation of some of the central ideas of ordinary differential equations such as the role of lipshitz conditions and compactness in existence theory. Students will be able to apply theory to the analysis of concrete model problems such as, for example, establishing existence of solutions of solutions including periodic solutions and establishing stability of solutions.

Contact and Advice

• The course coordinator is Dr Min-Chun Hong, room 549 in the Priestley Building, (building 67). If you have any comments or suggestions on the course or have questions on the course material, contact the coordinator by phone on 334 69036 or by email at mailto:hong@maths.uq.edu.au. You are welcome to ask any questions about the course during consultation hours, 11:00-12:00am on Wednesady. If you have questions about your current or future program of study, contact the chief academic advisor, honours advisor or postgraduate advisor .

Assumed Background

• The formal prerequisites are (MP202 or 212 or 272 or 282) + (MP203 or 213 or 273). Specifically, based on these courses students should be able to solve first and second order differential equations, have a basic appreciation of the general solution of linear systems, and have a basic knowledge of the classification of singular points for two dimensional first order systems.
• It is a student's own responsibility to fill in any gaps in their assumed knowledge. You may need to undertake background reading to understand the lecture material.

Teaching Mode

• Three hours of lectures and one hour of tutorial per week.
• Students are expected to attend lectures and tutorials.
• In tutorials students will be expected to attempt problems
• Students may work individually or in groups in tutorials
• Lecture : room 67-343, Tuesday 11-12;
• Lecture: room 67-342, Thursday 10-11;
• Lecture: room 67-343, Friday 11-12;
• Tutorial: room 67-641, Thursday 10-11
• There are no tutorials in week 1

Syllabus

• Statement of the initial value problem (IVP) for first order systems and equivalence with a system of integral equations. Background analysis including normed spaces and the Banach Contraction Mapping Principle.
• Existence for the IVP with lipschitz right side. Gronwall's inequality and variants. Differentiability with respect to parameters.
• Extendability of solutions.
• Linear systems, variation of parameters. Constant coefficient systems and Jordan forms. Linearised stability.
• Stability by Lyapunov's direct method.
• Periodic solutions. First integrals, the Morse Theorem and Attractors.
• Two dimensional homogeneous systems: phase plane analysis.
• Poincare-Bendixson Theory.
• Periodic solutions via Brouwer's Fixed Point Theorem and
• Degree Theory.
• Floquet's Theorem
• Perturbation Theory and the Poincare--Lindstedt method for periodic solutions.

Information Changes

• Any changes to course information will be announced in lectures and the information will be reproduced on the web page ( http://www.maths.uq.edu.au/courses/MATH4402). It is your responsibility to keep up to date with all information presented in your lecture group.

Resources

• References:
  1. Ferdnand Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer Verlag, New York, 1990
  2. E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955.
  3. P. Hartman, Ordinary Differential Equations, NYU, New York, 1967.
  4. J. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.
  5. W. Hurewicz, Lectures on Ordinary Differential Equations, John Wiley, New York, 1958.
  6. David A. Sanchez, Ordinary Differential Equations and Statility Theory: an Introduction, W.H. Freeman, San Francisco, 1968.
  7. Lawrence Perko, Differential Equations and Dynamcal Systems, Springer Verlag, New York, 1991.
  8. I.G. Petrovski, Ordinary Differential Equations, Prentice Hall, New York, 1966.
  9. R.L. Devaney, Introduction to Chaotic Dynamical Systems.
• Web: The course web page is at http://www.maths.uq.edu.au/courses/MATH4402. Information about the course and other resources are available there.

Assessment

• Assessment Scheme: There are two parts: the assignments counting 35% of the assessment and a final two hour examination counting 65% of the assessment.
• Final Examination: The final exam is closed book 2 hours long. It is worth 65%. Calculators without ASCII capabilities are permitted.
• Assignments: Four assignments are worth 35% of the final assessment. Each counting 35/400 of the assessment.
• Submission of assignments: Assignments must be submitted in your tutorial session.
  Dates for submission are
  Assignment 1: 17th March
  Assignment 2: 7th April
  Assignment 3: 18th May
  Assignment 4: 4th June
  Marks are awarded for correctness and clarity of presentation. One bonus mark per assignment may be awarded for innovative solutions. Part marks will be awarded for the successful completion of subtasks.
• If you miss an assessment item: In case of illness (or bereavement) you may be exempted from an assignment if a medical certificate (or other documentation) is received by the course co-ordinator within one week of the due date of the assignment. If you are exempted, then your assignment marks are weighted on a pro-rata basis. Note that ad hoc excuses (car trouble and the like!) will not be accepted; only documentation in connection with illness or bereavement . If you enrolled late then exemption will automatically be granted for anything missed before the date of enrolment.
• Missed assessment items: Failure to complete any item of assessment will result in a mark for zero for that component.
• Plagiarism: Plagiarism is the act of using other author's ideas and words or solutions without acknowledgement. It is a form of cheating and is considered as misconduct under official university policy and may attract severe penalties. You must not engage in plagiarism in any of your assignments. For more information, consult the libarary UseIt on Plagiarism: http://www.library.uq.edu.au/ssah/useits/plaguseit.html.
• Feedback on Assessment: Assistance for students, or The student Liason Officer, visit http://spider.sps.uq.edu.au/course_profile_info.pdf
• Assessment Criteria:
  • To earn a Grade of 7, a st udent must achieve a final mark of at least 85% and demonstr ate an excellent understanding of all of the theory. This includes clear expression of nearly all their deductions and explanations, the use of appropriate and efficient mathematical techniques and accurate answers to nearly all questions and tasks with appropriate justification.
  • To earn a Grade of 6, a student must achieve a final mark of at least 75% and demonstrate a comprehensive understanding of the theory of advanced analysis. This includes clear expression of most of their deductions and explanations, the general use of appropriate and efficient mathematical techniques and accurate answers to most questions and tasks with appropriate justification.
  • To earn a Grade of 5, a student must achieve a final mark of at least 65% and demonstrate an adequate understanding of the course theory. This includes clear expression of some of their deductions and explanations, the use of appropriate and efficient mathematical techniques in some situations and accurate answers to some questions and tasks with appropriate justification.
  • To earn a Grade of 4, a student must achieve a final mark of at least 50% and demonstrate an understanding of the basic concepts of advanced analysi s. This includes occasionally expressing their deductions and explanations clearly, the occasional use of appropriate and efficient mathematical techniques and accurate answers to a few questions and tasks with appropriate just ification. They will have demonstrated knowledge of techniques used to solve problems and applied this knowledge in some cases.
  • To earn a Grade of 3, a student must achieve a final mark of at least 45% and demonstrate some knowledge of the basic concepts of advanced analysis. This includes occasional expression of their deductions and explanations, the use of a few appropriate and efficient mathematical techniques and attempts to answer a few questions and tasks accurately and with appropriate justification. They will have demonstrated knowledge of techniques used to solve problems.
  • To earn a Grade of 2, a student must achieve a final mark of at least 20% and demonstrate some knowledge of the basic concepts of advanced analysis. This includes attempts at expressing their deductions and explanations and attempts to answer a few questions accurately.
  • A student will receive a Grade of 1 if they achieve a final mark of at most 19% or demonstrate extremely poor knowledge of the basic concepts in the course material. This includes attempts at answering some questions but showing an extremely poor understanding of the key concepts.

Personal Situation

• Disabilities: Any student with a disability who may require alternative academic arrangements in the course is encouraged to seek advice at the commencement of the semester from a Disability Adviser at Student Support Services.
• Personal Crises: If you feel that problems in your life are interfering with your university performance, you should consult Student Support Services in the Student Union complex. They offer academic and personal support in a confidential environment for free to students. They can provide a letter to give to lecturers which will help your case for a special exam (there is no longer special consideration for examinations), but do not leave it too late. If you are concerned about privacy with regard to medical certificates, please contact the University Health Service. With your permission, the Director will contact your treating practitioner to clarify the extent of your medical condition or other incapacity, and provide lecturers with a report - the Director is bound by confidentiality obligations. In any case, we prefer this course of action, as we are not qualified to assess medical evidence. The procedure outlined here accords with the University's policy on student privacy and confidentiality.