The ITN contracted 15 Early Stage Researchers (ESRs) to carry out the work programme at the main beneficiaries of this grant. The training of these ESRs is central to the project through the following means: (i) training at the host, (ii) training through secondments, (iii) training through network-wide events, in two different categories: (a) training through research, (b) training through courses, seminars and workshops. The ITN has held schools and workshops in the UK, Denmark, the Netherlands, Italy, Spain, Germany, Luxembourg and Ireland, and the ESRS and their supervisors participated in many externally organised international conferences and schools.
This first work package of the ITN concerned the development of bifurcation theory in new directions that are highly relevant to complex systems. In low-dimensional autonomous (time-independent) dynamical systems, critical transitions have been extensively studied as the discipline of bifurcation theory, explaining and classifying ways in which attractors lose stability and give rise to new types of behaviours. Central to these classical studies is the assumption of very slow (adiabatic) variation of parameters, but many complex systems deal with transitions in time-dependent (non-autonomous) systems, and often models have network or multiscale structure and are subjected to noise. The mathematical development of bifurcation theory for these classes of systems shapes the research themes of our first work package: bifurcations in nonautonomous dynamical systems, random dynamical systems, networks and multiscale systems. The ESRs have been extending various methodologies for local and global bifurcations in non-autonomous dynamical systems, in the context of rate-induced tipping, skew-product flows and Carathéodory (discontinuous) differential equations. Bifurcations in random dynamical systems are understood for certain low-dimensional systems, but are largely unexplored in higher dimensions. The ESRs have studied dimensional reduction techniques as well as non-chaotic attractors and stochastic models for the Atlantic Meridional Overturning Circulation. Some of the ESRs have studied critical transitions in random dynamical systems that have also a deterministic nonautonomous dependence: in the context of stochastic approximation algorithms, with applications to evolutionary game theory, critical transitions in the climate system such as transitions from the last glacial to the present warm interglacial and the cause of the abrupt Pleistocene ice age cycles, modelling paleo-climate via delay equations as well as the scaling behaviour in asset price bubbles in financial markets.
The second work package of the ITN aimed at the design of computational predictive tools and observables that can serve as early-warning signals for imminent critical transitions, validated by the theoretical results obtained in the research in the first work package. It is a central objective in many contexts, not just to understand how transitions happen, but also to predict them. In addition, it is of central importance whether any obtained early-warning signals are robust, in the sense that their validity extends over large classes of models and processes. The mathematical development of early-warning signals follows on the back of the theoretical developments in the first work package. First results concerns results on slow recovery rates signalling a smooth saddle node bifurcation, the use of extended centre manifolds to achieve early-warning signals in asset and bond price models and limitations of traditional early-warning signals for transitions into ice ages. Finally, several computational challenge have been addressed: numerical methods for the tracking of attractors of systems with bounded noise, numerical implementations of Lin’s method to study rate-induced tipping and simulations of so-called double-gyre flows modelling critical transitions in atmospheric circulation.
