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EPSRC Reference: EP/S005293/2
Title: Random Periodicity in Dynamics with Uncertainty
Principal Investigator: Zhao, Professor H
Other Investigators:
Researcher Co-Investigators:
Project Partners:
University of Warwick
Department: Mathematical Sciences
Organisation: Durham, University of
Scheme: EPSRC Fellowship
Starts: 01 August 2020 Ends: 31 December 2023 Value (£): 817,931
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
Statistics & Appl. Probability
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
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Summary on Grant Application Form
Random periodicity is ubiquitous in the real world. Examples include daily temperature variations, economic cycles, internet traffic volume and the activity of sunspots. However, we have little understanding of the dynamics of such systems. The lack of a mathematical model of random periodicity hinders scientific progress and development of applications e.g. in studying climate change and economic instability where randomness, nonlinearity and periodicity are present and interweave. It is now timely to build a theory of periodic random dynamical systems (PeriRDS) to remedy this.

I have introduced concepts of random periodic paths and periodic measures and provided a series of stochastic/functional analytic/numerical tools. The definition has been adopted by the random dynamical systems community especially in the study of stochastic bifurcations, random attractors, stochastic resonance, random horseshoe and in modelling real life problems such as climate dynamics. This research endeavour is rapidly evolving and many key results are missing. The random periodic scenario is excluded from the current ergodic theory even in the case of Markov chains. Its connection with the existence of pure imaginary eigenvalues of differential operators is a significant new observation but the associated spectral analysis still does not exist. Analysis of random periodicity in climate dynamics, computer networking and economics is still not available. Optimal control along random periodic paths is relevant theoretically and matters to applications, e.g. economic planning. So a formulation of such a mathematical model is critically important. Random periodic patterns can be found in many real world data sets. However, a random periodic model for forecasting future events and setting long term optimal control strategies based on these data, whilst crucial to applications, is still lacking.

The aim is to build a theory of periodic stochastic dynamics addressing all the above questions and to create a random periodic model for real world situations. Moreover, in many of these there is ambiguity, and the correct model for them requires the use of non-linear expectations. This non-classical probability theory is not well developed and even the theory for equilibria requires further fundamental work, though the Fellow and his group have made some progress. For this reason the second part of the programme will bring this theory to a state where it can also apply to periodic phenomena. This will be done by first getting a thorough understanding of the simpler phenomenon of equilibria in this context.
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