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Dynamical Systems
Provider: Faculty of Science

Activity no.: 5613-20-07-31 
Enrollment deadline: 20/04/2020
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time20.04.2020, at: 09:00 - 19.06.2020, at: 16:00
Regular seats50
ECTS credits7.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserChuang Xu    E-mail address: chuang@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Semester/BlockBlock 4
Scheme groupA (Tues 8-12 + Thurs 8-17)
Exam formWritten examination
Exam formWritten examination
Exam detailsWritten examination, 3 hours To qualify for the exam two written assignments must be approved (passed/not passed)
Course workload
Course workload categoryHours
Lectures28.00
Theory exercises14.00
Exam3.00
Preparation161.00
Sum206.00
Content
The course is an introduction to dynamical systems with emphasis on Ordinary Differential Equations (ODEs). The following topics will be discussed:
- Linear Systems: Existence and Uniqueness Theorem, Continuous Dependence on Initial Conditions and Parameters
- Stability Theory: Matrix solutions, Lyapunov functions
- Global Theory of Nonlinear Systems: Hartman-Grobman Theorem, Poincare-Bendixson Theorem
- Bifurcation theory: One and two dimensional bifurcations
- Discrete Dynamical Systems: Period doubling bifurcation
- Applications to chemistry, ecology, epidemiology and mechanics.

Learning outcome
Knowlegde:
Basic concepts in dynamic system theory: solutions, stability, bifurcation.
Standard methodes to determine the behavior of a dynamical system: Lyapunov function approach, eigenvalues of coefficient matrix
Local versus global behavior.

Skills:
Solve 2- and 3-dimensional linear ODEs with constant coefficients
Determine uniform / asymptotic stability of equilibria for linear systems
Prove stability / instability of gradient systems by constructing Lyapunov functions
Compute stable / unstable manifolds of simple 2-3 dimensional nonlinear systems
Identify fixed-point bifurcations and Hopf bifurcation for 1 or 2-dimensional ODEs
Describe bifurcations for simple discrete mappings

Competences:
Apply dynamical systems theory to build models and understand natural phenomena (for instance, Hopf bifurcation)

Literature
See Absalon for final course literature. The following is an example of expected course literature

EA Coddington and N. Levinson, Theory of ordinary differential equations. Tata McGraw-Hill Education, 1955.

L. Perko, Differential Equations and Dynamical Systems. 3rd Ed, Springer-Verlag, 2001.

MW Hirsch, S. Smale, RL Devaney, Differential Equations, Dynamic Systems and an Introduction to Chaos. Elsevier, Amsterdam, 2004.

Teaching and learning methods
4 hours lectures and 2 hours exercises per week for 7 weeks.

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