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Results regarding probable bifurcations from fixed points are presented inthe context of general dynamical systems (real, random matrices), time-delaydynamical systems (companion matrices), and a set of mappings known for theirproperties as universal approximators (neural networks). The eigenvalue spectrais considered both numerically and analytically using previous work of Edelmanet. al. Based upon the numerical evidence, various conjectures are presented.The conclusion is that in many circumstances, most bifurcations from fixedpoints of large dynamical systems will be due to complex eigenvalues.Nevertheless, surprising situations are presented for which the aforementionedconclusion is not general, e.g. real random matrices with Gaussian elementswith a large positive mean and finite variance.Comment: 21 pages, 19 figure

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective