Analysis of the transition between neurological states, such as a transition to a seizure-like state in people with epilepsy, is crucial to understanding and effective treatment of neurological disorders. Research into these transitions, also called transient dynamics, is an emerging area of interdisciplinary research that presents many challenges to study. We investigate transient dynamics in mathematical models of excitable cells, such as neurones, that comprise networks of coupled noisy dynamic nodes.
The well known Lorenz system is a simplified model of thermal atmospheric convection introduced by meteorologist Edward Lorenz in 1963. It is a parameter-dependent system of three ordinary differential equations. Warwick Tucker proved in 1999 that chaos exists in the system for the classic parameter values used by Lorenz. The analytic techniques used by Tucker and others to study the system rely on the condition that there is a stable foliation of the return map. This, in turn, means that the system can be reduced to a one-dimensional map, called the Lorenz map. We study a parameter regime where the foliation condition fails and much remains unknown about the system, including the parameter value at which the foliation condition itself is lost. Specifically, at how two-dimensional surfaces and one-dimensional curves, called invariant manifolds, organise the three-dimensional phase space of the Lorenz system and give structure to the chaos.
- JC, P Ashwin, and K Tsaneva-Atanasova, Sequential noise-induced escapes for oscillatory network dynamics, preprint.
- P Ashwin, JC and K Tsaneva-Atanasova, Fast and slow domino effects in transient network dynamics, arXiv:1701.06148.
- JC, B Krauskopf and H M Osinga, Finding first foliation tangencies in the Lorenz system, preprint.
- JC, B Krauskopf and H M Osinga, α-flips and T-points in the Lorenz system, Nonlinearity 28(3): R39–R65, 2015.