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 SaddleLoop Bifurcation p < pSL p = pSL p > pSL SaddleNode Invariant Circle p < pSNIC p = pSNIC p > pSNIC 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 l3 < l2 < 0 < l1 s = l1+ l2 s< 0: one stable limit cycle s> 0: one unstable limit cycle Saddle-Focus Re(l2,3) < 0 < l1 s = l1 + Re(l2,3) s < 0: one stable limit cycle s > 0: infinite # unstable limit cycles plus a stable chaotic attractor One-parameter Bifurcation Diagram SL SL HB Variable, x sss uss sss HB uss SN sss SN Parameter, p One-parameter Bifurcation Diagram SL uss Variable, x sss uss sss SNIC Parameter, p sss 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
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