In this article we investigate a pitchfork bifurcation of the local attractor of a simple capsizing model proposed by Thompson. Although this is a very simple system it has a very complicate dynamic. We try to reveal some properties of this dynamic with modern numerical methods. For this reason we approximate stable and unstable manifolds which connect the steady states to obtain a complete understanding of the topology in the phase space. We also consider approximations of the Lyapunov Exponents (resp. Floquet Exponents) which indicates the pitchfork bifurcation.

Macrophages are the largest cells in the immune system and they regulate inflammatory signalling and inform cell fate decisions. Many signals, including those mediated by Tumor Necrosis Factor alpha (TNFα) converge on a few key intracellular signalling pathways, including the Nuclear Factor kappa B (NFκB) network. The NFκB signalling pathway plays a vital role in the regulation of many different cellular responses, including the production of TNFα itself, which is required to sustain and propagate immune responses to, for example, infection or tissue damage. In this thesis we report on studies-both experimental and theoretical-of the NFκB signalling pathway in macrophages. Our collaborators stimulated these cells with various doses of Lipopolysaccharide (LPS), a molecule that forms the major component of the outer membrane of Gram-negative bacteria: in these experiments it serves as a proxy for bacterial infection. The macrophages, studied in vitro, respond as they are believed to do in tissues, by secreting certain signalling molecules called cytokines: the level of secretion proved to depend on the strength of the LPS stimulus. Further, heterogeneity of macrophage signalling was observed in response to a range of LPS doses. Within individual macrophages LPS stimulation results in oscillatory behaviour of NFκB localisation-NFκB shuttles in and out of the nucleus-with an amplitude (peak nuclear concentration) that also depends on the LPS dose. Heterogeneity was also observed in cells that were stimulated with the same dose intensity. This raises an important question about how immune cells coordinate inflammatory activity in the presence of this variability. In this thesis we aim to achieve an understanding of the system through the qualitative analysis of mathematical models of it. This work explores both the parametric sensitivity and bifurcation analyses for two mathematical models of NFκB in macrophages. Parametric sensitivity analysis is used to investigate the role of parameters on the model's output, especially on certain features of the signal-peak amplitudes, inter-peak intervals and areas beneath curves-that are commonly measured in single-cell experiments. Local bifurcation analysis is conducted in order to show all the possible behaviours produced when varying parameters.

In this article we investigate a pitchfork bifurcation of the local attractor of a simple capsizing model proposed by Thompson. Although this is a very simple system it has a very complicate dynamic. We try to reveal some properties of this dynamic with modern numerical methods. For this reason we approximate stable and unstable manifolds which connect the steady states to obtain a complete understanding of the topology in the phase space. We also consider approximations of the Lyapunov Exponents (resp. Floquet Exponents) which indicates the pitchfork bifurcation.