MATH759 INTRODUCTION TO CHAOTIC DYNAMICAL SYSTEMS
| Course Code: | 2360759 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Masters |
| Course Coordinator: | Prof.Dr. MARAT AKHMET |
| Offered Semester: | Fall or Spring Semesters. |
Course Objectives
This course serves as a direct continuation of the dynamical systems course and complements several other courses on differential and discrete equations. It provides a clear and informative introduction to the study of these topics. Together with the courses on dynamical systems and chaotic dynamics, it presents a comprehensive overview of the current state of theory in dynamical systems. Chaos theory is a crucial area of study at the forefront of differential equations, discrete equations, and their applications. It plays a significant role in various fields, including brain activity, neural networks, artificial intelligence, economics, finance, weather prediction, climate change, and many other research areas. The main educational objectives of this course are to prepare students from the mathematics department, as well as other departments at the university, for contemporary theoretical and practical areas of dynamical complexity. Students will be equipped to tackle the associated challenges in science, industry, and technology. The educational integrity of this program will be upheld through the creation and structure of the course.
Course Content
Mapping and Orbits. Periodic orbits, Linear mappings. Differentiable mappings. Hyberbolicity. Family of mappings Bifurcations. The quadratic map, Symbolic dynamics. Ingredients of chaos: Sensitivity, Dense orbits and transitivity. Schwarzian derivatives, Wiggly iterates. Cantor set and chaos. Higher dimensional dynamics: linear maps. The horseshoe map. Various routes to chaos.
Course Learning Outcomes
By the end of the course, the student will learn the following components of the theory:
- Mappings and Orbits. Periodic orbits. Linear mappings. Differentiable mappings
- Parametric family of mappings
- Bifurcation as a change of topological structure
- Period-doubling bifurcation
- Transitivity, Sensitivity, Density of periodic points
- One-dimensional chaos. The logistic map. Period-three points and chaos
- Method of bifurcation diagrams.
- Positive Lyapunov exponents.
- Alpha unpredictability and ultra Poincaré chaos.
- Chaos in stochastic processes.
- Two-dimensional maps: the Henon map; the horseshoe map
- Hyperbolicity. Homoclinic chaos.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution