# Hopf & other bifurcations

Ulf R. Andrick andrick at sun.rhrk.uni-kl.de
Thu Apr 30 14:58:28 EST 1992

(Sorry, if this appears a second time; but I have some problems with the
news servers here, and I never know if my articles are really sent or not.
If you have seen the first posting, then reply to that one, if possible.)

I thought the following request could be suitable for the newly-established
newsgroup sci.systems, but as there might be few readers in this group, yet,
I cross-post this article to some other groups which could be frequented
by people capable of giving some advice. Follow-ups are directed to
sci.systems.

Reading about neurone models I came across some analyses of model neurones
by means of non-linear dynamics. I learnt that Hodgkin-Huxley-type models
have a steady-state current-voltage curve consisting of two regions of
stable critical points joined by a region of unstable points. At the junctions
of these regions, Hopf bifurcations occur [1].

I also read that `a fixed point undergoes Andronov-Hopf bifurcation when
the Jacobian matrix of the system has the sum of the eigenvalues equal
to zero.'4
[2]

Now, I am only a poor, lowly student of biology, who did not encounter
non-linear
dynamics in his study before. I have some idea that the Jacobian matrix
describes
the changes of a multi-dimensional system, being sort of tangent to the curve.
But I cannot grasp what consequences some statements on the eigenvalues have,
such as the sum of them being zero or at least one of them being smaller than
zero or complex or whatever.

I can only remember dimly from school that with two-dimensional affinities, the
eigenvectors form the base of the transformed co-ordinate system
and that the eigenvectors are the corresponding scaling factors.

I do not have the time to learn all of the mathematics involved, but I would
like to have at least some, say, intuitive understanding of what the
eigenvalues
mean, esp. when complex. Is there any textbook which could shed some light
into my darkness without my having to study mathematics for years?

References:

[1] Vinet & Roberge, J. Theor. Biol. 147, 377 (1990)
[2] Borisyuk & Kirillov, Biol. Cybern. 66, 319 (1992)

--

Ulf R. Andrick                                    andrick at rhrk.uni-kl.de
Tierphysiologie
FB Biologie
Universitaet
D-W 6750 Kaiserslautern

Ulf R. Andrick                                  andrick at rhrk.uni-kl.de
Tierphysiologie
FB Biologie
Universitaet
D-W 6750 Kaiserslautern