MAT6019 Dynamic SystemsBahçeşehir UniversityDegree Programs MATHEMATICS (TURKISH, PHD)General Information For StudentsDiploma SupplementErasmus Policy StatementNational Qualifications
PhD. TR-NQF-HE: Level 8 QF-EHEA: Third Cycle EQF-LLL: Level 8

Course Introduction and Application Information

Course Code Course Name Semester Theoretical Practical Credit ECTS
MAT6019 Dynamic Systems Fall 3 0 3 8
This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester.

Basic information

Language of instruction: Turkish
Type of course: Departmental Elective
Course Level:
Mode of Delivery: Face to face
Course Coordinator : Prof. Dr. CANAN ÇELİK KARAASLANLI
Recommended Optional Program Components: Matlab
Course Objectives: The aim of this course is to convey the required concepts and skills to design, model, and simulate dynamic systems.

Learning Outcomes

The students who have succeeded in this course;
The students who succeded in this course:
will be able to:
Develop dynamical systems to model problems from biology, physics, and other areas.
Analyze dynamical systems; apply computer methods to solve and visualize more complex systems.
Determine the long term behavior of dynamical systems.
Set up, analyze, and interpret phase portraits of linear and non-linear systems of differential equations
Use graphical and symbolic methods to represent and interpret chaotic dynamical systems.

Course Content

Autonomous equations and systems ( Function spaces and orbits, critical points and linearization, Liouville theorem), Linear and non-linear systems and their critical points in two dimensions. Periodic solutions (Bendixon Condition, Poincare-Bendixon theorem). Stability (Stability of equilibrium solutions and periodic solutions). Linear equations ( Linear equations of constant and periodic coefficients). Bifurcation Theory (Center manifold, Normal forms and Local bifurcations)

Weekly Detailed Course Contents

Week Subject Related Preparation
1) Higher dimensions: The Lorenz system and chaos
1) Introduction to modeling and simulation
2) Linear dynamic systems; discrete and continuous time.
3) Nonlinear systems: fixed points, stability and linearization.
4) Lyapunov functions
5) Periodicity and Chaos
6) The Poincare-Bendixon Theorem
7) Hopf bifurcation
8) Periodicity in discrete time and stability of periodic points.
9) Nonlinear Techniques : Hamiltonian systems
10) Closed orbits and limit sets
11) Dynamical systems from biology.
12) Applications in Mechanics, Conservative systems.
14) The Lorenz system


Course Notes / Textbooks: 1- Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second Edition (Pure and Applied Mathematics) by Stephen Smale, Morris W. Hirsch and Robert L. Devaney (Nov 5, 2003)

2-Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity)
by Steven H. Strogatz.

3-Differential Equations and Dynamical Systems (Second Edition) by Lawrence Perko, published by Springer (1996).

Evaluation System

Semester Requirements Number of Activities Level of Contribution
Homework Assignments 3 % 10
Midterms 1 % 40
Final 1 % 50
Total % 100
Total % 100

ECTS / Workload Table

Activities Number of Activities Duration (Hours) Workload
Course Hours 14 3 42
Study Hours Out of Class 3 20 60
Homework Assignments 3 15 45
Midterms 1 23 23
Final 1 30 30
Total Workload 200

Contribution of Learning Outcomes to Programme Outcomes

No Effect 1 Lowest 2 Low 3 Average 4 High 5 Highest
Program Outcomes Level of Contribution