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Low-dimensional and Non-autonomous Dynamics

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New insight into non-autonomous dynamical systems

Complex dynamical systems ranging from climate change to financial markets can have ‘tipping points’ known as catastrophic bifurcations, where a sudden change in a condition can cause the system’s equilibrium to be lost. Detailed studies on the near-bifurcation points can help control bifurcation properties and achieve in a way the desired system’s dynamical behaviour.

Fundamental Research

Modelling an incredible range of behaviours such as planet motion in the solar system or disease spread in a population, dynamical systems are commonly used for system analysis in a wide range of scientific fields. Until now, several local bifurcations (including saddle-node, transcritical, pitchfork, period-doubling and Hopf bifurcations) have been studied in physics, biology, engineering, ecology and economics. Despite progress in the field, further understanding of bifurcations in non-autonomous dynamical systems is still needed. The EU-funded project LDNAD (Low-dimensional and non-autonomous dynamics) was established to provide new insight and tools to complement the study of non-autonomous bifurcation theory, especially of the non-autonomous counterparts of the classical bifurcation pattern of dynamical systems. Research into non-autonomous Hopf bifurcations focused on a long-standing problem in the field concerning the usual Hopf bifurcation pattern that gives way to the two-step scenario for the non-autonomous Hopf bifurcation proposed by Ludwig Arnold. Researchers described this scenario either by deterministically forced models, which can be treated as continuous skew product systems on a compact product space, or randomly forced systems that lead to skew products over a measure-preserving base transformation. In this scenario, external forcing can lead to a separation of the complex conjugate eigenvalues, giving rise to the two-step bifurcation scenario, in which an invariant ‘torus’ splits off a previously stable central manifold. Researchers proved that this torus consists of a topological circle in each fibre. The project team also noted significant progress in issues related to the ergodic theory. Considering that the lexicographic order induces a partial order – known as first-order stochastic dominance on the collection of its shift-invariant probability measures, researchers studied the fine structure of this dominance order and proved that the Sturmian invariant measures are totally ordered with respect to this order. LDNAD worked to increase understanding of bifurcations in non-autonomous dynamical systems – namely the effect of changes occurring in the structure of dynamical systems when parameters are varied.

Keywords

Dynamical systems, non-autonomous dynamics, LDNAD, bifurcation theory, ergodic theory

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