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Nonautonomous Bifurcation Theory

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Small changes can redefine entire systems

Dynamical systems are used to detail a wide variety of phenomena ranging from the simple movements of a pendulum through biological interactions to sociological patterns. A dynamical system is subject to various rules governing the motions of its application processes.

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Real life situations can be described by applying the mathematical concept of dynamical systems, which depend on factors that reflect various conditions influencing the system. Examples of non-autonomous situations found in the applied sciences include climate modelling and studies of pollution-spreading processes. The dynamical bifurcation theory tackles the qualitative change in behaviour of a system whose parameters have been altered. The 'Nonautonomous bifurcation theory' (NBT) project sought to develop the bifurcation theory of non-autonomous systems beyond the traditional setting. The EU-funded project set a number of objectives to further work in this area. These included the characterisation of bifurcations of low-dimensional non-autonomous dynamical systems, studying higher-dimensional bifurcation scenarios, developing a bifurcation theory for random systems with bounded noise, and developing a concept of topological equivalence for non-autonomous dynamical systems. Researchers were successful in classifying bifurcations in one-dimensional non-autonomous differential equations and characterising discontinuous bifurcations in random systems with bounded noise. Team members developed a numerical scheme for detecting controlled heteroclinic orbits (a path that joins two different equilibrium points). This can be applied to a model for ship roll-motion. Other achievements include developing a concept of exponential attractivity and bifurcation for finite-time systems and a numerical scheme for approximating invariant sets in random systems with bounded noise. Advances in knowledge made during the NBT project contributed to how the concept of a non-autonomous dynamical system can be applied to model real world phenomena.

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