A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 5: Two-Parameter Bifurcation Diagrams

1. A Primer in BifurcationTheoryfor Computational Cell BiologistsLecture 5: Two-Parameter Bifurcation Diagrams 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. HB Variable, x SN Parameter, p Variable, x SN Parameter, p One-Parameter Bifurcation Diagrams

3. y x One-Parameter Bifurcation Diagrams Variable, x Parameter, p

4. y x One-Parameter Bifurcation Diagrams Variable, x Parameter, p

5. y x One-Parameter Bifurcation Diagrams Variable, x Parameter, p

6. One-Parameter Bifurcation Diagrams Variable, x Parameter, p

7. = 0 Numerical Bifurcation Theory Two equations in three unknowns. Fix p = po; solve for (xo, yo). Expand using Taylor’s Theorem:

8. This is perfectly generalizable to any number of variables. As long as With this equation, we can follow a steady state as p changes. As we follow a locus of steady states, we can monitor Whenever D= 0, we have located a SN bifurcation point, and whenever R = 0, we have located a Hopf bifurcation.

9. HB SN SN Variable, x Variable, x Parameter, p Parameter, p

10. cusp point one ss Parameter, q one ss fold line three ss Parameter, p Two-parameter Bifurcation Diagram Three equations in four unknowns. Fix p = po; solve for (xo, yo, qo). Follow the solution using… D = det(J) “codimension-two” “codimension-one”

11. s s sxs Variable, x Parameter, p Two-parameter Bifurcation Diagram Parameter, q Parameter, p

12. s s Variable, x sxs Parameter, p Two-parameter Bifurcation Diagram Parameter, q Parameter, p

13. Variable, x Parameter, p Two-parameter Bifurcation Diagram “universal unfolding” s s Parameter, q sxs Parameter, p “codimension-two”

14. x p

15. x q p

16. HIGH x LOW q p

17. x Medium x p q High x Low x p q Bistable (High or Low)

18. Variable, x Parameter, p Two-parameter Bifurcation Diagram Three equations in four unknowns. Fix p = po; solve for (xo, yo, qo). Follow the solution using… one sss R = Re(l1,2) Parameter, q one uss one slc one sss Parameter, p

19. Sub-critical Hopf Bifurcation CF supHB subHB Variable, x Parameter, p

20. s u s Variable, x Parameter, p Two-parameter Bifurcation Diagram degenerate HB supHB CF Parameter, q subHB degenerate HB Parameter, p