FYS-3035 Stochastic modelling and non-linear dynamics - 10 stp

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.

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

Lectures: 40 hours

Exercises: 24 hours