A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 4: Global Bifurcations

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A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 4: Global Bifurcations. http://www.biology.vt.edu/faculty/tyson/lectures.php. John J. Tyson Virginia Polytechnic Institute &amp; Virginia Bioinformatics Institute. Click on icon to start audio. Signal-Response Curve =

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A Primer in BifurcationTheoryfor Computational Cell BiologistsLecture 4: Global Bifurcations

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

John J. Tyson

Virginia Polytechnic Institute

& Virginia Bioinformatics Institute

Click on icon to start audio

Signal-Response Curve =

One-parameter Bifurcation Diagram

• Hopf Bifurcation (oscillations)
• Subcritical Hopf
• Cyclic Fold

Homoclinic Orbits

Heteroclinic Orbits

p < pHC

p = pHC

p > pHC

Heteroclinic Orbits

p < pSL

p < pSNIC

p = pSL

p = pSNIC

p > pSL

p > pSNIC

Homoclinic Orbits

Loop

Bifurcation

Node

Invariant

Circle

Homoclinic Bifurcation

Finite amplitude, small frequency, infinite period

Hopf Bifurcation

Small amplitude, frequency = Im(l), finite period

Andronov-Leontovich Theorem

In a two-dimensional system, a homoclinic orbit gives birth to a finite amplitude, large-period limit cycle; either stable:

or unstable:

• > 0: one unstable limit cycle s > 0: infinite # unstable limit cycles

plus a stable chaotic attractor

Shil’nikov Theorem

In a three-dimensional system, a homoclinic orbit gives birth to a stable or unstable limit cycle, or to much more complicated behavior …

l3 < l2 < 0 < l1 Re(l2,3) < 0 < l1

s = l1+ l2 s = l1 + Re(l2,3)

SL

SL

sss

uss

sss

uss

sss

SN

SN

HB

HB

One-parameter Bifurcation Diagram

Variable, x

Parameter, p

One-parameter Bifurcation Diagram

SL

sss

uss

sss

uss

Variable, x

sss

SNIC

Parameter, p

References
• Strogatz, Nonlinear Dynamics and Chaos (Addison Wesley)
• Kuznetsov, Elements of Applied Bifurcation Theory (Springer)
• XPP-AUT www.math.pitt.edu/~bard/xpp
• Oscill8 http://oscill8.sourceforge.net