Grant: Discovery Grants Program - Individual

We will use methods from dynamical systems to explore the behaviour of mathematical models of differentially heated rotating fluid systems, as well as of systems from other areas such as neuroscience. In so doing, we will develop a better understanding of the fundamental nature of these systems. Specific objectives include analyzing a model for the differentially heated rotating fluid annulus experiment to gain insight into certain experimentally observed phenomena. Two examples are (1) the triple-point, which is a point in the space of control parameters that is shared by the boundaries of three dynamically distinct flow regimes, and (2) the vacillation flow pattern, which is a wave-like motion where the amplitude or wave structure appear to vary periodically in time. The triple-point will be investigated by quantitatively evaluating a strongly resonant Hopf bifurcation, while the nature of vacillation will be investigated by studying the secondary bifurcations that occur in the model. Also, we will investigate the importance of the spatial extent of the dendrites in determining the behaviour of a neuron by studying the dynamics of a multi-compartment neuronal model near a codimension-3 degenerate Bogdanov-Takens bifurcation. The proposed research is important to the field of applied dynamical systems, because, among other reasons, (1) the bifurcations of codimension 2 and 3 that we will study, including degenerate and resonant bifurcations, and bifurcations with symmetry, are not only of inherent interest due to their fundamental role in characterizing the dynamics of a system, but are also of interest because they are being studied in the context of scientifically relevant models, and (2) the numerical approximations that must be made are challenging, some of which are at the limit of computational feasibility. The proposed research is also of direct scientific importance because the analysis will isolate the relevant nonlinear interactions that occur in the phenomena of interest, and thus will provide insight into the important mechanisms that lead to the observed behaviour.