Maximal delay in stability loss phenomenon
Description: This is a project in the field of perturbation theory for dynamical systems. Its objective is to provide a systematic asymptotic description of the maximal delay of stability loss (known as a buffer point) phenomenon at passage through a bifurcation for systems with slowly varying parameters. This is a challenging open problem in perturbation theory of dynamical systems deeply connected with other scientific fields such as mechanics (e.g., flutter phenomenon), physics (e.g., operation of argon lasers), chemistry (e.g., Belousov-Zhabotinsky reaction with varying parameters), and biology (e.g., neuronal bursting).
In classical bifurcation theory, the behaviour of systems depending on a parameter is considered for values of this parameter close to some critical, bifurcational one. This is a static theory in the sense that the parameter value does not vary with time. In theory of dynamical bifurcations, the parameter changes slowly over time and passes through values that would be bifurcation points in the classical static theory. A standard general framework for description of dynamics here is that of slow-fast dynamical systems. These systems have variables of two types: fast and slow ones. Slow variables represent slowly varying parameters. Some phenomena arising here are drastically different from seemingly reasonable predictions based on static theory. Stability loss delay is one of such phenomena. Assume that at some value of a parameter an equilibrium of a system for fast variables for frozen values of parameters (a fast system) loses its asymptotic linear stability while remaining non-degenerate. It turns out that in analytic systems stability loss delays are inevitable: phase points attracted to the stable equilibrium of the fast system remain near the unstable equilibrium for a long time after the bifurcation, during which the parameters change by a quantity of order 1. Such delay generally does not exist in non-analytic (even infinitely smooth) systems.
The existence of a maximal delay is a fundamental property of the stability loss delay phenomenon in generic analytic slow-fast systems. Its manifestation is that all the phase points that are attracted to stable equilibrium prior to some threshold value of a parameter, leave a neighbourhood of unstable equilibrium almost simultaneously near some other threshold value of a parameter. Stability loss delay beyond this parameter value (the buffer point) is not possible unless the initial data are of a very specific form. These two new threshold parameter values are related to singularities of solutions in complex time, a relationship that is still to be fully understood and represents one of the main challenges in this area.
Amount: £495,100
Funding type: research
Start date: 2026-02-01
End date: 2029-02-01
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Funded outputs count: 1
Funder award ID: UKRI1724
OpenAlex ID: G851369080