Dynamical Systems and Chaos (MATH467) Course Detail

Course Name	Course Code	Season	Lecture Hours	Application Hours	Lab Hours	Credit	ECTS
Dynamical Systems and Chaos	MATH467	Area Elective	4	0	0	4	6

Pre-requisite Course(s)
N/A

Course Language	English
Course Type	Elective Courses
Course Level	Natural & Applied Sciences Master's Degree
Mode of Delivery	Face To Face
Learning and Teaching Strategies	Question and Answer.
Course Coordinator
Course Lecturer(s)
Course Assistants
Course Objectives	The mathematical formulation of numerous physical problems results in differential equations which are actually nonlinear. This course is about dynamical aspects of nonlinear ordinary differential equations. It treates chiefly autonomous systems, emphasizing qualitative behavior of solution curves, and gives an introduction to the phase portrait analysis of such systems.
Course Learning Outcomes	The students who succeeded in this course; be able to learn the existence theorem and learn continuation of solution and dependence on initial value. learn the linear systems, how to diagonal a matrix and how to use it. learn how to obtain the exponentials of operators, solve Linear Systems in R^2 by using the Eigenvalues. use Jordan forms, learn stability theory and learn application of these on nonhomogeneous linear Systems. find the maximal interval of existence, learn the flow that a differential equation defines and learn the linearization. classify the equilibrium points, stable and center manifold theory, stability and Liapunov functions learn definitions and applications of limit sets, attractors, Hamiltonian systems, the Poincare-Bendixson theory and bifurcation theory.
Course Content	One-dimensional dynamic systems, stability of equilibria, bifurcation, linear systems and their stability, two-dimensional dynamic systems, Liapunov?s direct method and method of linearization, 3-dimensional dynamic systems.

Weekly Subjects and Releated Preparation Studies

Week	Subjects	Preparation
1	Linear Systems: Uncoupled Linear Systems, Diagonalization	pp. 1-6
2	Exponentials of Operators, The Fundamental Theorem for Linear Systems, Linear Systems in R^2	pp. 6-20
3	Complex Eigenvalues, Multiple Eigenvalues	pp. 20-32
4	Jordan Forms, Stability Theory, Nonhomogeneous Linear Systems	pp. 32-64
5	Nonlinear Systems: Some Preliminary Concepts and Definitions, The Fundamental Existence-Uniqueness Theorem, Dependence on Initial Conditions and Parameters	pp. 65-79
6	The Maximal Interval of Existence, The Flow Defined by a Differential Equation, Linearization	pp. 79-105
7	Midterm
8	The Stable Manifold Theorem, Stability and Liapunov Functions	pp. 105-119 and pp. 129-136
9	Saddles, Nodes, Foci and Centers, Nonhyperbolic Critical Points in R^2, Center Manifold Theory	pp. 136-163
10	Nonlinear Systems: Global theory, dynamical systems and global eExistence theorems, limit sets and attractors	pp. 181-202
11	Periodic Orbits, Limit Cycles, The Stable Manifold Theorem for Periodic Orbits	pp. 202-211 and pp. 220-234
12	Hamiltonian Systems, The Poincare-Bendixson Theory in R^2, Bendixson's Criteria	pp. 234-252 and pp. 264-267
13	Nonlinear Systems: Bifurcation Theory, Structural Stability	pp. 315-334
14	Bifurcations at Nonhyperbolic Equilibrium Points	pp. 334-343
15	Review
16	Final Exam

Sources

Course Book	1. L. Perko, Differential Equations and Dynamical Systems: 3rd Edition, Springer, New York, 2000.
Other Sources	2. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems: 2nd Edition, Springer, New York, 1996.
	3. M.W. Hirsch, S. Smale and R.L. Devaney, Differential Equations, Dynamical Systems and, An Introduction to Chaos: 2nd Edition, Academic Press, San Diego, 2004.
	4. W. Kelley and A.Peterson, The Theory of Differential Equations: Classical and Qualitative, Pearson Education, New Jersey, 2004.
	5. S.L.Ross, Differential Equations, 3rd edition, Wiley, New York, 1984

Evaluation System

Requirements	Number	Percentage of Grade
Attendance/Participation	-	-
Laboratory	-	-
Application	-	-
Field Work	-	-
Special Course Internship	-	-
Quizzes/Studio Critics	-	-
Homework Assignments	2	20
Presentation	-	-
Project	-	-
Report	-	-
Seminar	-	-
Midterms Exams/Midterms Jury	1	40
Final Exam/Final Jury	1	40
Toplam	4	100

Percentage of Semester Work	60
Percentage of Final Work	40
Total	100

Course Category

Core Courses	X
Major Area Courses
Supportive Courses
Media and Managment Skills Courses
Transferable Skill Courses

The Relation Between Course Learning Competencies and Program Qualifications

#	Program Qualifications / Competencies	Level of Contribution
#	Program Qualifications / Competencies	1	2	3	4	5
1	Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2	Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3	Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4	Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5	Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6	Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7	Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8	Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9	Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10	Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11	Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

ECTS/Workload Table

Activities	Number	Duration (Hours)	Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class	16	4	64
Presentation/Seminar Prepration
Project
Report
Homework Assignments	2	8	16
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury	1	16	16
Prepration of Final Exams/Final Jury
Total Workload			96