Stochastic modelling and non-linear dynamics
About the course
The course is available as a singular course. The course is also available to exchange students and Fulbright-students.
The course will only be taught if there is a sufficient number of students. If you are interested in following the course, please contact the student advisor as soon as possible
The course introduces stochastic modelling both as a method for simplified descriptions of complex systems and as realistic models for describing and understanding statistical properties of data time series. These properties include probability distributions, auto-correlation functions, frequency spectra and extreme event statistics.
Students will use case studies of non-linear, chaotic, and turbulent deterministic systems to learn about instabilities, transitions from laminar to chaotic and turbulent states, and the effects of non-linearities on the evolution of complex systems. Stochastic modelling will be used to aid in the understanding of such systems. Numerical computations, model simulations and data analysis are central in the course.
The models considered include filtered Poisson processes stochastic differential equations, the Van der Pol and Lotka-Volterra oscillators, the chaotic Lorenz, Rössler and logistic map systems, as well as Rayleigh-Benard convection and the Kuramoto-Sivashinsky equation.
Part of the course content is dynamic and will reflect the interests of the participating students and the ongoing research.
Admission requirements
Objectives of the course
Knowledge - The student can:
- identify stationarity in time series and assess options for stationarizing time series
- describe the filtered Poisson process, explain its statistical properties, and quantify the concept of intermittency
- describe routes to chaos in deterministic systems
- describe the transition from laminar to turbulent convection
- describe how symmetries in dynamical equations relate to symmetries in the statistical properties chaotic or turbulent solutions
- give examples of how chaotic and stochastic systems have been used to aid understanding of experimental and simulation data from complex systems such as fusion and astrophysical plasmas, neutral fluids, population dynamics and the global climate
Skills - The student can:
- analyze and interpret statistical properties of time series data from experimental measurements and numerical simulations
- identify relevant stochastic models for a given data set based on its statistical properties
- solve basic analytical problems concerning filtered Poisson processes and stochastic differential equations
- make realizations of filtered Poisson processes using both the direct summation and convolution methods
- make realizations of stochastic differential equations using the Millstein and second-order stochastic Runge-Kutta methods
- perform linear stability analysis of ordinary and partial differential equations and analyze mode structures
- perform numerical simulations and time-series analysis of non-linear dynamical systems using filtered Poisson process and stochastic differential equations
General competence - The student can:
- understand applications and limitations of linear modelling and assumed Gaussian statistics
- perform data analysis of strongly non-Gaussian process in the Python programming language
Prerequisites
Recommended prerequisites
FYS-1001 Mechanics, FYS-2006 Signal processing, MAT-2200 Differential Equations, MAT-2201 Numerical Methods, STA-2003 Time series
Credit reduction
If you pass the examination in this course, you will get an reduction in credits (as stated below), if you previously have passed the following courses:
- FYS-8035 Stochastic modelling and non-linear dynamics 8 ects
Teaching methods
Lectures: 40 hours
Exercises: 24 hours
Language of instruction and examination
The language of instruction is English and all of the syllabus material is in English. Examination questions will be given in English, but may be answered either in English or a Scandinavian language.Schedule
The schedules are normally finalized and published well in advance of the start of the semester, often a few weeks beforehand. This gives students the opportunity to organize their studies and prepare for upcoming activities.
It is recommended to check the schedule regularly, as changes may occur.
Information to incoming exchange students
This module is open for exchange students with a Bachelor's degree in physics or similar education.
Do you have questions about this module? Please check the following website to contact the course coordinator for exchange students at the faculty: https://en.uit.no/education/art?p_document_id=510412
Examination
| Oral exam | Duration: 45 Minutes |
Grade: A–E, fail F |
To take an examination, the student must have passed the following coursework requirements
| Mandatory Assignment 1 | Grade: Approved – not approved |
| Mandatory Assignment 2 | Grade: Approved – not approved |
Everything you need to know about before, during, and after the exam; registration, absence, appeals, and diplomas: UiT Exams homepage
More info about the coursework requirements
To take an examination, the student must have passed the following coursework requirements:
2 mandatory assignments, pass/fail