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

1. 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

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

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

4. p < pHC p = pHC p > pHC Heteroclinic Orbits

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

6. Homoclinic Bifurcation Finite amplitude, small frequency, infinite period Hopf Bifurcation Small amplitude, frequency = Im(l), finite period

7. 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:

8. < 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 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)

9. SL SL sss uss sss uss sss SN SN HB HB One-parameter Bifurcation Diagram Variable, x Parameter, p

10. One-parameter Bifurcation Diagram SL sss uss sss uss Variable, x sss SNIC Parameter, p

11. 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