Scaling of Saddle-Node Bifurcations: Degeneracies and Rapid Quantitative Changes

Details Scaling of Saddle-Node Bifurcations: Degeneracies and Rapid Quantitative Changes Scaling of Saddle-Node Bifurcations: Degeneracies and Rapid Quantitative Changes

2008-07-09

arxiv.org/abs/0807.1546v2

Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 4, (045101) pp. 1-9, 2009

Mathematics

Dynamical Systems

preprint version - for final version see journal reference

Scientific paper

10.1088/1751-8113/42/4/045101

The scaling of the time delay near a "bottleneck" of a generic saddle-node bifurcation is well-known to be given by an inverse square-root law. We extend the analysis to several non-generic cases for smooth vector fields. We proceed to investigate $C^0$ vector fields. Our main result is a new phenomenon in two-parameter families having a saddle-node bifurcation upon changing the first parameter. We find distinct scalings for different values of the second parameter ranging from power laws with exponents in (0,1) to scalings given by O(1). We illustrate this rapid quantitative change of the scaling law by a an overdamped pendulum with varying length.

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