John J. Tyson Virginia Polytechnic Institute

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Presentation transcript:

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

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

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

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

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

x1 x1 f

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

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

HB Torus CASE 3 SN HB SN

Heteroclinic x1 x1

Torus

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

CASE 4 From Kuznetsov’s Book

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

CASE 1 From Kuznetsov’s Book

CASE 2

CASE 3