John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute

1. A Primer in BifurcationTheoryfor Computational Cell BiologistsLecture 6: Takens-Bogdanov Bifurcation http://www.biology.vt.edu/faculty/tyson/lectures.php John J. Tyson Virginia Polytechnic Institute & Virginia Bioinformatics Institute Click on icon to start audio

2. Variable, x Parameter, p Cusp Bifurcation “universal unfolding” s s Parameter, q sxs Parameter, p

3. u s Variable, x Parameter, p Degenerate Hopf Bifurcation “universal unfolding” supHB s CF Parameter, q subHB Parameter, p

4. uxs s s s s x sxs s x Variable, x s xs u ulc s Parameter, p Takens-Bogdanov Bifurcation subHB Parameter, q “universal unfolding” SL SN Parameter, p

5. osc s s Parameter, q sxs Parameter, p Bistability & Oscillations in Chemical Reactors inflow stirrer Pacault, Vidal, deKepper, Boissonade 1970’s, CNRS, Bordeaux France outflow “Cross-shaped Phase Diagram”

6. l k SN subHB Toy Model Guckenheimer (1986) Physica D 20:1-20

8. u Variable, x SLC Parameter, p Saddle-Node Loop Bifurcation SN u xs Parameter, q uxs SL SNIC u x s Parameter, p

9. u u Variable, x xs Parameter, p Saddle-Node Loop & Takens-Bogdanov Bifurcations cusp s s HB sxs Parameter, q SL TB SNL SNIC s uxs SN Parameter, p

10. Neutral Saddle-Loop Bifurcation CF SL SL