1. A Primer in Bifurcation Theory for Computational Cell Biologists Lecture 3: Hopf Bifurcation
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. Stability Analysis of Steady States
and eigenvectors of the Jacobian matrix.

4. If det(J) = 0, then l = 0 is an eigenvalue, and
the steady state is a saddle-node.
p < pSN
p = pSN
p > pSN
det(J) = 0

5. Hopf Bifurcation
If J has a pair of complex conjugate eigenvalues,
l = Re(l) ± i Im(l), and Re(l) = 0, then the steady
state is undergoing a Hopf bifurcation.
Super-critical
Hopf bifurcation
Sub-critical
Hopf bifurcation

6. The Hopf Bifurcation Theorem
The Hopf Bifurcation Theorem. Suppose that at p = pH the Jacobian matrix has a pair of complex conjugate eigenvalues, l = Re(l) ± i Im(l), with Re(l) = 0 and dRe(l)/dp > 0. Then, for |p - pH| sufficiently small, there exists a one-parameter family of small amplitude limit cycles for one of the following three conditions:
(1) p > pH, in which case the limit cycles are stable.
(2) p < pH, in which case the limit cycles are unstable.
(3) p = pH, in which case the limit cycles are neutral.
Comments. Case (3) is unusual (‘structurally unstable’). In cases (1) and (2), the limit cycles are parameterized by the distance from the bifurcation point: |p - pH|. The amplitude of the limit cycle grows like SQRT(|p – pH|), and the period of the limit cycle is close to 2p/Im(l).

7. x x p p Supercritical Hopf Bifurcation sss uss max min
unstable
limit
cycle
Subcritical Hopf Bifurcation
max x
sss
uss
min
stable
limit
cycle p

8. Numerical Bifurcation Theory
How to locate a Hopf bifurcation?
When following a steady state as p changes, simply look for
Re(l) = 0
How to follow a periodic solution? 1
t = t/T
x(t)
Hopf’s theorem tells us how to approximate the periodic solution for |p – pH| sufficiently small in terms of sin(wt) and cos(wt), where w = Im(l). Use this initial guess to compute the exact periodic solution and then follow the solution as p changes.

9. Stability of Periodic Solutions
The stability of a periodic solution is determined by its “Floquet Multipliers”. Let P be a plane that is transverse to the periodic solution at a particular point: P
Let y0 = x(0)-x0 be a small initial displacement in P from the periodic solution at x0. Starting at y0, integrate the ODEs forward in time until you return to P at point y1 = x(T)-x0. This procedure defines a nonlinear mapping yk+1 = Y(yk), which can be…

10. (See Kuznetsov, Section 5.3)
Re(m)
Im(m)
…linearized: yk+1 = Myk .
The eigenvalues mi of the matrix M are known as the Floquet multipliers of the periodic solution. The periodic solution is stable if all the multipliers lie within the unit circle in the complex plane
Period-doubling
bifurcation
Torus
Cyclic-fold
(See Kuznetsov, Section 5.3)

11. A typical bifurcation diagram
slc 1 5
slc
ulc
Period-doubling 3
Cyclic Fold
ulc
uss x
sss
Subcritical Hopf Bifurcation 4 2 p

12. References Strogatz, Nonlinear Dynamics and Chaos (Addison Wesley)
Kuznetsov, Elements of Applied Bifurcation Theory (Springer)
XPP-AUT
Oscill8