A Primer in Bifurcation Theoryfor Computational Cell Biologists

Lecture 4: Global Bifurcations

John J. TysonVirginia Polytechnic Institute

& Virginia Bioinformatics Institute

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

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•Saddle-Node (bistability, hysteresis)•Hopf Bifurcation (oscillations)•Subcritical Hopf•Cyclic Fold•Saddle-Loop•Saddle-Node Invariant Circle

Signal-Response Curve = One-parameter Bifurcation Diagram

Homoclinic Orbits Heteroclinic Orbits

saddle-loop

saddle-node-loop

saddle-saddle-connection

Heteroclinic Orbits

p = pHC

p < pHC

p > pHC

Homoclinic Orbits

p = pSLp < pSL p > pSL

p = pSNICp < pSNIC p > pSNIC

Saddle-Loop

Bifurcation

Saddle-Node

InvariantCircle

Hopf Bifurcation

Homoclinic Bifurcation

Finite amplitude, small frequency, infinite period

Small amplitude, frequency = Im(), 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:

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 …

Saddle Saddle-Focus

3 < 2 < 0 < 1 Re(2,3) < 0 < 1

= 1+ 2 = 1 + Re(2,3)

< 0: one stable limit cycle < 0: one stable limit cycle> 0: one unstable limit cycle > 0: infinite # unstable limit cycles

plus a stable chaotic attractor

One-parameter Bifurcation Diagram

Parameter, p

Var

iabl

e, x

sss

uss

uss ssssss

SN

SN

HB

HB

SL

SL

One-parameter Bifurcation Diagram

Parameter, p

Var

iabl

e, x

sss

uss

uss ssssss

SL

SNIC

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