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A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 7: Fold-Hopf Bifurcation

A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 7: Fold-Hopf 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. degenerate Hopf. cusp.

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A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 7: Fold-Hopf Bifurcation

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  1. A Primer in BifurcationTheoryfor Computational Cell BiologistsLecture 7: Fold-Hopf 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. degenerate Hopf cusp supHB q s s s CF sxs q xs u uxs s p u subHB u s p subHB Saddle- Node Loop SN s xs q sxs s Takens- Bogdanov q SL SL uxs SNIC SN p p Codimension-Two Bifurcations

  3. HB p2 SL SN p1 Takens-Bogdanov Bifurcations x1 p2 saddle-loop p1

  4. Hopf SN 4 p2 3 1 2 p1 Fold-Hopf Bifurcation x1 p2 p1

  5. r x1 constant angular velocity in f x2 x1 x3 Minimum number of variables for fold-Hopf bifurcation is three:

  6. x1 x1 f

  7. (− + +) (− − −) r (+ − −) (+ + +) x1 x1 p1 HB HB SN SN HB CASE 1 SN HB SN

  8. (− − −) (− + +) x1 (+ + +) (+ − −) p1 HB HB SN SN HB CASE 2 SN r x1 HB SN

  9. HB Torus CASE 3 SN HB SN

  10. Heteroclinic x1 x1

  11. Torus

  12. x1 p1 HB He To HB SN SN HB Torus CASE 3 SN Heteroclinic HB SN

  13. CASE 4 From Kuznetsov’s Book

  14. CASE 4 x1 HB HB To SN SN p1 ‘Cycle Blowup’

  15. CASE 1 From Kuznetsov’s Book

  16. CASE 2

  17. CASE 3

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