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MiS Preprint

Probability of local bifurcation type from a fixed point: a random matrix perspective

David Albers and J. Sprott


Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance.

MSC Codes:
37-XX, 34-XX
05.45.-a, 89.75.-k
bifurcaton theory, random matrices, high-dimensional dynamical systems

Related publications

2006 Repository Open Access
David J. Albers and J. C. Sprott

Probability of local bifurcation type from a fixed point: a random matrix perspective

In: Journal of statistical physics, 125 (2006) 4, pp. 889-925