# 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 4 0 0 4 6
Pre-requisite Course(s)
Math 231 (Linear Algebra I) or Math 275 (Linear Algebra) and Math 262 (Ordinary Differential Equations)
Course Language English N/A Bachelor’s Degree (First Cycle) Face To Face Question and Answer. 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. 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. 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. 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

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 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
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 Can apply gained knowledge and problem solving abilities in inter-disciplinary research
3 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
4 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
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility
7 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
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2)
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach.
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.

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