A broadly applicable method for the identification of bifurcations in complex, multi-parameter dynamical systems

By Daniel K Wells1, Adilson Motter1, William L Kath1

1. Northwestern University

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Abstract

Traditional numerical bifurcation methods seek to identify and classify solutions in systems with small numbers of parameters. These methods have become canonical tools in determining how solutions of dynamical systems depend upon parameters. Of particular interest are bifurcation points, which are points in parameter space at which different solution branches intersect. Difficulties arise when such methods are applied to models with many parameters, however, since in these cases it is unclear which parameters or combinations of parameters should be changed to locate bifurcation points. Here we introduce a method based on the Wentzell-Freidlin action to locate bifurcation points to a prescribed precision in large, multi-parameter dynamical systems. We first illustrate the method on simple examples with only a few parameters, and then apply it to induce a bifurcation that eliminates the cancerous state in a 54-dimensional, 193-parameter model of large granular lymphocyte leukemia. This approach is highly scalable and broadly applicable to a wide class of dynamical systems.

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