When a slow-moving system crosses a tipping point, it often clings to its old behaviour for far longer than expected—and this project aims to understand exactly how long that delay can last. This research tackles a fundamental gap in the mathematics of dynamical systems. Classical bifurcation theory assumes parameters are fixed, but in the real world—from aircraft wings to neurons—parameters drift slowly. When a system loses stability, it can linger near the unstable state for a surprisingly long time before jumping to a new behaviour. This "stability loss delay" is well known, but the *maximum possible* delay—the so-called buffer point—remains poorly understood. The project will develop a systematic mathematical description of this maximal delay, linking it to hidden singularities in complex time. This is primarily curiosity-driven fundamental science. There are no immediate practical applications. However, the mathematics developed here could eventually help engineers predict when a flutter in a wing or a surge in a laser will become unavoidable, or help biologists understand the precise timing of neuronal bursts. Past work on similar delay phenomena has already reshaped how we model climate tipping points and chemical oscillations. A deeper understanding of maximal delay could, in time, sharpen predictions in any system where slow change meets sudden collapse.
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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.
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