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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

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

slide2

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

slide3

HB

p2

SL

SN

p1

Takens-Bogdanov Bifurcations

x1

p2

saddle-loop

p1

slide4

Hopf

SN

4

p2

3

1

2

p1

Fold-Hopf Bifurcation

x1

p2

p1

slide5

r

x1

constant angular velocity in f

x2

x1

x3

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

slide6

x1

x1

f

slide7

(− + +)

(− − −)

r

(+ − −)

(+ + +)

x1

x1

p1

HB

HB

SN

SN

HB

CASE 1

SN

HB

SN

slide8

(− − −)

(− + +)

x1

(+ + +)

(+ − −)

p1

HB

HB

SN

SN

HB

CASE 2

SN

r

x1

HB

SN

slide9

HB

Torus

CASE 3

SN

HB

SN

slide12

x1

p1

HB

He

To

HB

SN

SN

HB

Torus

CASE 3

SN

Heteroclinic

HB

SN

slide13

CASE 4

From Kuznetsov’s Book

slide14

CASE 4

x1

HB

HB

To

SN

SN

p1

‘Cycle

Blowup’

slide15

CASE 1

From Kuznetsov’s Book