• To be able to solve the linear and nonlinear differential equation system
• Understanding the stability of the equation system
• Learning basic and important theorems for dynamical systems
• Learning Bifurcation theory
20
Contribution of the Course to Professional Development
fixed points, stability, Lyapunov functions.
Stability analysis, potential function, bifurcation in one dimensional autonomous systems,
Linear autonomous systems and Lyapunov functions for them, stability and Lyapunov functions
Nonlinear autonomous systems, local analysis at fixed points, nonlinear centers, conserved systems, reversible systems
• Index theory, Limit cycles, Dulac criterion, orbital stability definition.
• Poincare-Bendixsion Theorem, Linard systems.
Hopf bifurcation
21
Learning Outcomes:
1
fixed points, stability, Lyapunov functions.;
2
Stability analysis, potential function, bifurcation in one dimensional autonomous systems,;
3
Linear autonomous systems and Lyapunov functions for them, stability and Lyapunov functions;
4
Nonlinear autonomous systems, local analysis at fixed points, nonlinear centers, conserved systems, reversible systems;
5
• Index theory, Limit cycles, Dulac criterion, orbital stability definition.;
6
• Poincare-Bendixsion Theorem, Linard systems.;
7
• Hopf bifurcation;
22
Course Content:
Week
Theoretical
Practical
1
Autonomous dynamical systems, existence and uniqueness, fixed points and stability.
2
Lyapunov functions. Stability analysis in one dimensional autonomous systems, potential function,
3
bifurcations in one-dimensional autonomous systems,
4
bifurcations
5
Stability in linear autonomous systems
6
Stability and Lyapunov functions, two-dimensional linear autonomous systems
7
Nonlinear autonomous systems, local analysis of fixed points, nonlinear centers
8
Conservative systems, reversible systems
9
Index theory
10
Limit cycles, Dulac criterion
11
Orbital stability definition, Poincare-Bendixson Theorem
12
Poincare-Bendixsion Theorem, Linard systems.
13
Hopf bifurcation
14
Hopf bifurcation
23
Textbooks, References and/or Other Materials:
• Perko,L.(2001). Differential Equations and Dynamical Systems, Springer. • Wiggins S. (2003). Introdution to Applied Nonlinear Dynamical Systems and Chaos, Addison-Wesley. • Lynch, S.(2010). Dynamical Systems with Applications Using MAPLE, İkinci Sürüm, Birkhauser. • Miller, R. K. ve Michel, A. N.(1982). Ordinary Differential Equations, Academic Press. • Cronin, J. (2008). Ordinary Differential Equations - Introduction and Qualitative Theory, Üçüncü Sürüm, CRC Press.
24
Assesment
TERM LEARNING ACTIVITIES
NUMBER
PERCENT
Midterm Exam
1
40
Quiz
0
0
Homeworks, Performances
0
0
Final Exam
1
60
Total
2
100
Contribution of Term (Year) Learning Activities to Success Grade
40
Contribution of Final Exam to Success Grade
60
Total
100
Measurement and Evaluation Techniques Used in the Course
Understanding the principles of applied mathematics used in the course
Information
25
ECTS / WORK LOAD TABLE
Activites
NUMBER
TIME [Hour]
Total WorkLoad [Hour]
Theoretical
14
2
28
Practicals/Labs
14
2
28
Self Study and Preparation
8
5
40
Homeworks, Performances
0
7
14
Projects
1
10
10
Field Studies
0
0
0
Midtermexams
1
15
15
Others
0
0
0
Final Exams
1
15
15
Total WorkLoad
150
Total workload/ 30 hr
5
ECTS Credit of the Course
5
26
CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS
