ME506 DYNAMICS OF NONLINEAR SYSTEMS
| Course Code: | 5690506 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mechanical Engineering |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Assist.Prof.Dr HAKAN ÇALIŞKAN |
| Offered Semester: | Fall and Spring Semesters. |
Course Objectives
The objective of this course is to equip students with modern tools to analyze and predict the behavior of nonlinear dynamical systems—oscillations, limit cycles, and stability—using time-domain (Lyapunov/LaSalle), frequency-domain (describing functions; Popov, circle/parabola criteria), and perturbation/approximation methods (multiple scales, Poincaré–Lindstedt, Krylov–Bogoliubov). Emphasis is on modeling, rigorous analysis, and simulation-driven insight for engineering systems.
Course Content
Introduction to nonlinear systems. Limit cycle analysis. Piecewise linearization. Forced nonlinear systems. Approximation methods; multiple time scale, Poincare perturbation, Krylov and Bogoliubov methods. Describing function analysis. Stability of nonlinear systems; Lyapunov theory, Aizerman's and Kalman's conjectures, Lure's problem, Popov, circle and parabola criteria.
Course Learning Outcomes
By the end, students will be able to:
- Model nonlinear systems and detect/interpret limit cycles (including via piecewise linearization).
- Apply multiple-scale and Poincaré perturbation methods to approximate periodic solutions.
- Use Krylov–Bogoliubov (KBM) averaging to derive amplitude/phase evolution for weakly nonlinear oscillators.
- Compute and use describing functions to predict existence/stability of oscillations in forced nonlinear feedback.
- Formulate Lur’e-type interconnections and assess absolute stability.
- Apply Lyapunov’s direct method estimate regions of attraction.
- Evaluate Aizerman’s and Kalman’s conjectures; recognize conditions and counterexamples.
- Use Popov, circle, and parabola criteria (and related positive-real/KYP viewpoints) for frequency-domain stability certification.
- Design and execute simulations (MATLAB) to validate analytical predictions; build Poincaré maps and phase portraits.
- Communicate analyses clearly in technical reports with assumptions, limitations, and verification.
Program Outcomes Matrix
| Contribution | |||||
| # | Program Outcomes | No | Yes | ||
| 1 | Acquires the fundamental scientific knowledge required to analyze and solve advanced-level problems in the field of mechanical engineering. | ✔ | |||
| 2 | Gains the competence to utilize advanced engineering mathematics methods in the formulation, analysis, and solution of engineering problems. | ✔ | |||
| 3 | Conducts literature reviews using printed and online sources, analyzes the collected literature, and identifies the current state-of-the-art in the relevant scientific field. | ✔ | |||
| 4 | Demonstrates the ability to prepare and deliver a seminar on a technical subject. | ✔ | |||
| 5 | Develops the ability to conduct independent research on a specific topic and solve advanced engineering problems. | ✔ | |||
| 6 | Contributes to the national and/or international body of knowledge through original research. | ✔ | |||
| 7 | Gains the competence to effectively communicate the process and results of research conducted on a specific subject through scientifically structured written reports and oral presentations. | ✔ | |||
| 8 | Acquires the ability to publish research findings as articles in national and/or international scientific journals and/or present them as papers at conferences. | ✔ | |||
| 9 | Acts in accordance with universal principles of research and publication ethics. | ✔ | |||