# TMA4165 Differential Equations and Dynamical Systems - Spring 2018

### Messages

**Takk for et hyggelig semester og god sommer!****Exam (9.8):**- The exam and solutions can be found on the previous exam page!
- The solutions present one way to solve the exam problems! If your solution differs it might still be ok!
- If you find a mistake / typo in the proposed solution, send an email!
- 13.8: Corrected a mistake in the solution to Problem 3b!

**Grading of the exam (18.6):**- I handed in your grades today, but it might take a few days until you get them.
- Here are some comments concerning your solutions.

**Exam (7.6):**- The exam and solutions can be found on the previous exams page!
- The solutions present one way to solve the exam problems! If your solution differs it might still be ok!
- If you find a mistake / typo in the proposed solution, send an email!
- 8.6: Corrected a small mistake in the solutions to Problem 2!

**Exam:****Final curriculum:**see further down.- Fractals and chaos is
**not**part of the curriculum this year

**Learning outcomes:**here**Support material code D:**- No printed or hand-written support material is allowed. A specific basic calculator is allowed.

- Since the course is taught in English, the exam will be given only in English. You are free to answer in Norwegian or English.
**Mattelab:**- 27.4: 8:00-10:00 A32
- 1.6: 13:00-15:00, S24
- 4.6: 13:00-15:00, S24

**Week 12 (19.-23.3):**- All questions can be answered with the material covered so far!
- Supervision: 19.3 (Exercise class)
- No lectures: 22.-23.3

**Tentative curriculum:**see further down!- Challenge:
- The Challenge is solved!
- If you gave it a try but didn't succeed, you can ask for a detailed explanation!

- Welcome to TMA4165 Differential Equations and Dynamical Systems!
**There have been some changes to the timetable!**- 1st lecture:
**Thursday 11.1.** - 1st exercise class: Monday 15.1

### General information

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

- Wikipedia (2017), read the full article here.

### Lecturer

- Office 1150, Sentralbygg 2
- Office hours: by appointment

### Lectures

- Thursday 14:00-16:00 S6
- Friday 08:00-10:00 A32 (Handelshøyskolen)

### Exercise assistant

### Exercises

- Monday 15:00-17:00 G1
- A problem set will be given each week.
- A project will be given at the beginning of the semester.
- Both the problems sets and the project will be supervised in the exercise class. They are part of the curriculum, but are not required to be handed in.
- In general the exercise class is the right place to
- get help with the problem sets (both part a and b, although the main focus is on part b)
- get help with the project
- discuss and ask questions about the material covered in the course

### Books and reading material

**Main book:**D.W. Jordan & P. Smith:*Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers.*Fourth edition, Oxford University Press 2007.**Note:**H. Hanche-Olsen: Assorted notes on dynamical systems**Note:**The project on Traveling wave solutions for the Korteweg-de Vries equation

**Further reading:**

- G.C. Layek:
*An Introduction to Dynamical Systems and Chaos,*Springer, New Delhi, 2015. - H. Logemann and E.P. Ryan:
*Ordinary Differential Equations - Analysis, Qualitative Theory and Control,*Springer Undergraduate Mathematics Series, Springer, London, 2014. - L. Perko:
*Differential Equations and Dynamical Systems,*Third Edition, Texts in Applied Mathematics 7, Springer-Verlag, New York, 2001. - I.R. Shafarevich and A.O. Remizov:
*Linear Algebra and Geometry,*Springer-Verlag, Berlin Heidelberg, 2013. (Covers e.g. Jordan normal forms in detail.) - G. Teschl:
*Ordinary Differential Equations and Dynamical Systems,*Graduate Studies in Mathematics, Volume 140, Amer. Math. Soc., Providence, 2012. (Covers e.g. matrix exponentials and Hartman-Grobman in detail.)

### Final curriculum

- J.S: Chap. 1, 2, 3 (3.3. and 3.5. excl.), 8, 10, 11.1-3, 12.1-4, 13.1
- H: Chap. 1, 2, 3, 4, 5, 6
- All exercises

### Reference group

- If you have any remarks, request, complaints etc. regarding this course, contact the lecturer or the reference group.

### Exam

- Dato: 7.6.2018
- Support material code D: No printed or hand-written support material is allowed. A specific basic calculator is allowed.
- If the course is taught in English, the exam will be given only in English. Students are free to answer in Norwegian or English.