Stochastic modelling and non-linear dynamics

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 are sufficiently many students. Please contact the student adviser as soon as possible if you are interested in following the course.

Programstudents may register for the course through Studentweb. The registration deadline is September 1st/February 1st.

Other PhD students at UiT and external applicants may apply for admission through Søknadsweb, application code 9301. The application deadline is June 1st for the autumn semester and December 1st for the spring semester

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.

PhD students or holders of a Norwegian master´s degree of five years or 3+ 2 years (or equivalent) may be admitted. PhD students must upload a document from their university stating that there are registered PhD students. This group of applicants does not have to prove English proficiency and are exempt from semester fee. Holders of a Master´s degree must upload a Master´s Diploma with Diploma Supplement / English PhD students at UiT The Arctic University of Norway register for the course through StudentWeb . External applicants apply for admission through SøknadsWeb. Application code 9303.

All external applicants have to attach a confirmation of their status as a PhD student from their home institution. Students who hold a Master of Science degree, but are not yet enrolled as a PhD-student have to attach a copy of their master's degree diploma. These students are also required to pay the semester fee.

Knowledge - The student can:

• describe and identify stationarity in time series and assess options for stationarizing time series
• describe the filtered Poisson process, derive its statistical properties, and quantify the concept of intermittency
• describe routes to chaos and turbulent states in deterministic systems including fluid flows and plasma dynamics
• describe how symmetries in dynamical equations relate to symmetries in the statistical properties of chaotic or turbulent solutions
• describe bifurcations and the transition from laminar to turbulent convection 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 and the global climate

Skills - The student can:

• analyze and interpret statistical properties of time series data from experimental measurements and numerical simulations
• Use conditional averaging to identify and extract coherent structures in time series data
• identify relevant stochastic models for a given data set based on its statistical properties
• solve advanced 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 analytical modelling and data analysis of strongly non-Gaussian process in the Python programming language

Anbefalte forkunnskaper

FYS-1001 Mechanics, FYS-2006 Signal processing, MAT-2200 Differential Equations, MAT-2201 Numerical Methods, STA-2003 Time series

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-3035 Stochastic modelling and non-linear dynamics 8 ects

Lectures: 40 hours

Exercises: 24 hours

Language of instruction and examination

Re-sit examination