# Seminars

These are research seminars run by the Mathematics of Life and Social Sciences Group.

## Seminar details

Day and time: Tuesdays from 2 pm - 3 pm.
Venue: 39 AA 04.
Open to: Staff and postgraduate research students.

## Upcoming seminars

Date: 7 Feb 2023

Title: A talk of two halves:  “Practical catastrophe theory”, and “Hidden dynamics of maps and sleep cycles”

Speaker: Dr Mike Jeffrey, University of Bristol

Abstract:

During covid I did the unusual thing of discovering something that might actually be useful. Actually two things, so I’d like to present them both briefly. One concerns how to handle discontinuities in maps, and one concerns how we locate bifurcations in (also unusually for me) smooth dynamical systems. I’ll give examples taken from reaction-diffusion PDEs in part 1, and from sleep-wake cycle maps in part 2.

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Part 1. Practical catastrophe theory for the modern age

Catastrophe theory over the last half century went from being sensational to controversial to obscure. It gave birth to the indispensable theories of bifurcations and singularities, but I want to show you how incredibly useless these theories are! (compared to what they should be, anyway). Then I’ll use a novel extension of Thom’s elementary catastrophes to unleash the full might of singularity theory for use on vector fields, on PDEs, and on other things they shouldn’t apply to. In the process we reduce vast numbers of calculations needed to find a singularity in standard theory (95 million to find a “butterfly” in 4 dimensions), to just a few to locate the “underlying catastrophe" (7 to find the “butterfly” in 4 dimensions).

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Part 2. Hidden dynamics of maps and sleep cycles, and when “period 1+2 implies chaos”

It is time to restore law and order to the study of “maps with discontinuities”. Most of the laws governing a map like  xn+1=f(xn) are broken if the map f has a discontinuity or ‘gap’. I’ll show you that the laws of continuous maps are largely restored once we include the “hidden dynamics” of the discontinuity. This hidden dynamics fills in the gaps in bifurcation diagrams, and restores the fundamental rule that “period 3 implies chaos”, but with a twist, that now  “period 1+2 implies chaos”.

Date: 21 February 2023
Title: TBA (Discussion)
Speaker: Prof. Lorenzo Fioramonti, Surrey Institute for Sustainability, University of Surrey, UK

Date: 7 March 2023
Title: Mathematical model of sexual response
Speaker: Dr Konstantin Blyuss, University of Sussex, UK

In this talk I will discuss a mathematical model of Masters-Johnson human sexual response cycle. As a starting point, I will review cusp catastrophe and will show why earlier studies that interpreted sexual response cycle using this catastrophe were incorrect. I will then present a derivation of a phenomenological psycho-physiological model of human sexual response cycle. Bifurcation analysis is performed to identify stability properties of the model’s steady state, and numerical simulations are performed to illustrated different types of dynamics associated with the cycle. We will then look at the stochastic version of the model, where I will discuss properties of the spectrum and variance of stochastic oscillations around deterministically stable steady state, as well as the computation of confidence regions. To make a better understanding of stochastic dynamics, I will show how large deviation theory can be used to compute optimal escape paths from the neighbourhood of the steady state, and will discuss clinical implications of results.

Date: 16 May 2023
Title: TBC
Speaker: Dr Tomas Diviak, University of Manchester, UK

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