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

D. J. Albers and J. C. Sprott

Departments of Physics, University of Wisconsin, 1150 University Avenue, Madison, WI 53706, USA

Received October 23, 2005; accepted October 9, 2006
Published Online: December 2, 2006


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 spectrum 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 does not hold, e.g., real random matrices with Gaussian elements with a large positive mean and finite variance.

Ref: D. J. Albers and J. C. Sprott, Journal of Statistical Physics 125, 889-925 (2006)

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