1. A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 7: Fold-Hopf Bifurcation
John J. Tyson
Virginia Polytechnic Institute
& Virginia Bioinformatics Institute
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2. Codimension-Two Bifurcations
supHB CF s u
subHB p
degenerate Hopf q p q s
sxs
cusp s
sxs
uxs
Takens-
Bogdanov p SN xs SL
subHB q p
uxs SL u xs SN
SNIC
Saddle-
Node
Loop q

3. Takens-Bogdanov Bifurcationsx1 p2
saddle-loop p1 SN SL HB p2 p1

4. Fold-Hopf Bifurcationx1 p2 p1 p1 p2 2 3 4 1 SN
Hopf

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

6. x1 x1 f

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

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

9. HB
Torus
CASE 3 SN HB SN

10. Heteroclinic x1 x1

11. Torus

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

13. CASE 4
From Kuznetsov’s Book

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

15. CASE 1
From Kuznetsov’s Book

16. CASE 2

17. CASE 3