# 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

### ABSTRACT

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)

The complete paper is available
in PDF format.

Return to Sprott's Books and Publications.