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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 6046 NON-LINEAR DIFFERANTIAL EQUATIONS AND DYNAMICAL SYSTEMS ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

DOCTOR MELTEM ALTUNKAYNAK

Offered to

Mathematics (English) Mathematics (English)

Course Objective

This course aims to present the classical theory of linear systems and the theory of nonlinear and chaotic systems with continous and discrete time systems.

Learning Outcomes of the Course Unit

1   will be able to obtain and classify the critical points of dynamical systems. 2   will be able to analyze the stability of a nonlinear dynamical system using both linearization and Liapunov functions. 3   will be able to analyze how to apply the Poincare-Bendixson theorem to determine the existence of a closed orbit. 4   will be able to apply techniques of dynamical systems to analyze solutions of problems from applied mathematics and other fields. 5   will be able to identify and classify bifurcations in nonlinear differential equations. 6   will be able to use perturbation techniques to determine approximate solutions of nonlinear differential equations.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Linear Theory of Dynamical Systems: Fundamental solutions, Autonomous Linear Systems and Phase Portraits, Critical Points and Stability 2 Nonlinear Theory of Dynamical Systems: Autonomous Nonlinear Systems and Phase Portraits, orbits and critical points 3 Linearization of nonlinear systems at a critical point 4 Stability by Linearisation, Asymtotic stability of trivial solution,instability of periodic solutions 5 Periodic solutions, stability of periodic solutions, Hamiltonian systems and systems with first integrals 6 Conservative force fields and elliptical planetary orbits, Hamiltonian Mechanics, 7 Volterra-Lotka Predator-Prey equations 8 Liapunov functions, Liapunov stability analysis 9 Introduction to perturbation theory,The Poincare expansion theorem 10 The Poincare-Lindstedt method 11 Bifurcation Theory , Center manifolds 12 Bifurcations of critical points and Hopf bifurcation 13 Chaos, The Lorenz equations 14 One dimensional chaos, Liapunov exponents, Review of the term

Recomended or Required Reading

Non-linear Differential Equations and Dynamical Systems, F. Verhulst, Springer - Verlag, 1989. Supplementary Book(s): Differential Equations, Dynamical Equations and Linear Algebra, M.W. Hirsch and S. Smale Dynamical Systems with Applications using Mathematica. Stephen Lynch. Invitation to Dynamical Systems. Edward R. Scheinerman

Planned Learning Activities and Teaching Methods

Lecture notes Problem Solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 ASG ASSIGNMENT 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE MTE * 0.30 +ASG * 0.20 +FIN * 0.50 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + ASG * 0.20 + RST * 0.50

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

meltem.topcuoglu@deu.edu.tr

Office Hours

Will be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 14 3 42 Preparations before/after weekly lectures 14 4 56 Preparation for midterm exam 1 25 25 Preparation for final exam 1 35 35 Preparing assignments 5 8 40 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 204

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 LO.1 5 5 LO.2 3 5 5 LO.3 5 5 LO.4 5 4 3 5 5 LO.5 4 4 5 LO.6 4 3 5