BioND — Dynamics of Biological Networks

Computation and visualization of bifurcation surfaces

Dirk Stiefs, Thilo Gross, Ralf Steuer, and Ulrike Feudel
International Journal Bifurcation and Chaos 18(8), 2191-2206, 2008.


Triangulation of a Hopf bifurcation surface

The localization of critical parameter sets called bifurcations is often a central task of the analysis of a nonlinear dynamical system. Bifurcations of codimension 1 that can be directly observed in nature and experiments form surfaces in three-dimensional parameter spaces. In this paper, we propose an algorithm that combines adaptive triangulation with the theory of complex systems to compute and visualize such bifurcation surfaces in a very efficient way. The visualization can enhance the qualitative understanding of a system. Moreover, it can help to quickly locate more complex bifurcation situations corresponding to bifurcations of higher codimension at the intersections of bifurcation surfaces. Together with the approach of generalized models the proposed algorithm enables us to gain extensive insights in the local and global dynamics not only in one special system but in whole classes of systems. To illustrate this ability we analyze three examples from different fields of science.