A Primer in Bifurcation Theoryfor Computational Cell BiologistsLecture 7: Fold-Hopf Bifurcation

John J. TysonVirginia Polytechnic Institute

& Virginia Bioinformatics Institute

http://www.biology.vt.edu/faculty/tyson/lectures.php

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Codimension-Two Bifurcations

p

qs

sxss

cusp supHB

CF

s

u

ssubHB

p

degenerate Hopf

q

s

sxs

uxs

s

Takens-Bogdanov

p

SN

s xs

SL

subHB

q

p

uxs

SL

u xs

u

SN

SNIC

Saddle-Node Loop

q

Takens-Bogdanov Bifurcations

1,2

1

Re ( , ) 0 (Hopf)

( , ) 0 (fold)

( , ) 0 (steady state)

x p

x p

f x p

p1

p2

x1

saddle-loop

p1

SN

SL

HB

p2

Fold-Hopf Bifurcation

1

2,3

( , ) 0 (steady state)

( , ) 0 (fold)

Re ( , ) 0 (Hopf)

f x p

x p

x p

p1

p2

x1

p1

p2

2

3

4

1

SNHopf

Minimum number of variables for fold-Hopf bifurcation is three:

1 2 3 1 2 3( , , ) ( , , ) where ix x x x x ix e

x1

constant angular velocity in

x1

x2

x3

x1

x1

SN

SN

HB

HB

x1

x1

p1SN SNHB HB

(+ − −)

(− − −)(− + +)

(+ + +)

CASE 1

SN

SN

HB

HB

x1

x1

p1SN SNHB HB

(+ − −)

(− − −) (− + +)

(+ + +)

CASE 2

SN

SN

HB

HB

CASE 3Torus

x1x1

Heteroclinic

Torus

SN

SN

HB

HB

CASE 3Torus

Heteroclinic

x1

p1SN SNHB HBToHe

CASE 4

From Kuznetsov’s Book

CASE 4

x1

p1

SN SNHB HB To

‘CycleBlowup’

CASE 1

From Kuznetsov’s Book

CASE 2

CASE 3