EPSRC DTP PhD studentship: Record events for chaotic dynamical systems

University of Exeter - College of Engineering, Mathematics and Physical Sciences

Main supervisor: Dr Mark Holland (University of Exeter)

Project Description:

This project analyses record events for chaotic dynamical systems. The scope lies at the interface of mathematical analysis, probability and statistics.

A challenging and active area in dynamical systems is that of return time statistics. Namely, given a dynamical system and a specific region of phase space, what is the probability distribution that governs the times of first return to this region? The proposed project will exploit recent developments on the theory of return time statistics to understand the statistics of record events for dynamical systems.  For a real valued time series sequence of observations on a dynamical system, a record event corresponds to the time for which this sequence attains its maximum (record) value. For chaotic systems, this project will develop a theory to understand the probability distributions that govern these record times, and corresponding record values.

For independent identically distributed (i.i.d) random variables the record time distribution is known, thus it is a pertinent question to study this problem for dynamical systems. In this case the random variables are now replaced by a stationary (discrete time) stochastic process generated by the time series of observations on the dynamical system. Since these observations are not independent the i.i.d theory cannot directly be applied and new ideas are needed.

This project will use approaches in mathematical analysis of dynamical systems, but there are clear applications in statistics. Given good progress, the project will explore prediction of records in weather events (e.g. occurrence of record temperatures), or financial events (e.g. occurrence of record market crashes).

Student's role: To commence the project the student will be introduced to the subject of dynamical systems/ergodic theory and be given a range of mathematical models to explore (analytically and numerically).This will include intermittent systems and chaotic systems where there is a vast array of literature and recent developments to the theory. The research novelty is to use dynamical system approaches to understand record statistics for dependent processes. Hence the student will also be expected to undertake research in probability theory. Given good progress the student will apply their theory to case studies, including predicting records in climate/weather and/or financial data. Thus there is opportunity for the student to engage with the statistical sciences in record prediction.

Within dynamical systems and ergodic theory, the student will be introduced to a wide team of national/international experts in the field via links of the supervisor (Dr Holland). This includes the LMS funded UK one-day ergodic theory meeting network which has regular meetings throughout the year, and through the supervisor's international collaborations: Universities in the USA, Netherlands, France, Portugal and Hamburg.

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South West England