Models of adaptation and behavioural patterns in complex systems in crisis

"Change is the only constant in life," said the Greek philosopher Heraclitus. In order to function and develop under uncertainty, our complex society, as well as its elementary agents, must exhibit an appropriate adaptation. Hence, it is imperative to understand how adaptation takes place in complex systems. This understanding should include a theory of adaptation that would enable us to model and predict its outcomes using the language of mathematics. In this work, we provide an attempt to present such a theory. A review of major directions is provided in Chapter 1 of the thesis. Chapter 2 provides a hierarchy of models of adaptation. These models are based on Hans Selye's and Bernard Goldstone's axioms of adaptation: an organism (or a group of individuals) is represented as a system which optimizes distribution of the internal adaptation resource ("Adaptation Energy") for neutralization of an aggressive factor. A general analysis of these models is presented in the same chapter. In this Chapter 3, we provide an example of the application of these principles and models to the problem of understanding complex spontaneous activity of neural cultures. We demonstrate that the rich dynamics of activity patterns in neural cultures can be described by very simple equations modelling "adaptation" in such systems. In Chapter 4 we present another take on adaptation through correlation graphs measuring correlations between agents in complex systems. We discuss a potential of using these graphs for early warning of crisis for various systems. We demonstrate this on examples of the thirty largest companies from the Financial Times Stock Exchange 100 Index during the two major crises - the global financial crisis of 2008 and the first wave of the COVID-19 pandemic in March 2020. Chapter 5 concludes the thesis and discusses directions for future work.