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 & Virginia Bioinformatics Institute Click on icon to start audio

Signal-Response Curve = One-parameter Bifurcation Diagram Saddle-Node (bistability, hysteresis) Hopf Bifurcation (oscillations) Subcritical Hopf Cyclic Fold Saddle-Loop Saddle-Node Invariant Circle

Homoclinic Orbits Heteroclinic Orbits saddle-loop saddle-saddle-connection saddle-node-loop

Heteroclinic Orbits p = pHC p < pHC p > pHC

Homoclinic Orbits p = pSL p < pSL p > pSL p = pSNIC p < pSNIC Saddle- Loop Bifurcation p = pSL p < pSL p > pSL Saddle- Node Invariant Circle p = pSNIC p < pSNIC p > pSNIC

Homoclinic Bifurcation Hopf Bifurcation Small amplitude, frequency = Im(l), finite period Homoclinic Bifurcation Finite amplitude, small frequency, infinite 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 l3 < l2 < 0 < l1 Re(l2,3) < 0 < l1 s = l1+ l2 s = l1 + Re(l2,3) < 0: one stable limit cycle s < 0: one stable limit cycle > 0: one unstable limit cycle s > 0: infinite # unstable limit cycles plus a stable chaotic attractor

One-parameter Bifurcation Diagram SL HB sss uss HB SN 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