1. A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 4: Global Bifurcations
John J. Tyson
Virginia Polytechnic Institute
& Virginia Bioinformatics Institute
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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. Heteroclinic Orbits
p = pHC
p < pHC
p > pHC

5. 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

6. Homoclinic Bifurcation
Hopf Bifurcation
Small amplitude, frequency = Im(l), finite period
Homoclinic Bifurcation
Finite amplitude, small frequency, infinite 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. 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 < l Re(l2,3) < 0 < l1
s = l1+ l 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

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

10. One-parameter Bifurcation DiagramSL
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
Oscill8