http://en.wikipedia.org/wiki/Hopf_bifurcation

These non-linear dynamic systems are pretty ubiquitous in nature. For example, they occur fairly naturally in 'neurocomputational' models of brain function -- see Eugene M Izhikevitch's wonderful webpage:

http://www.izhikevich.org/human_brain_simulation/Blue_Brain.htm

... including many publications: http://www.izhikevich.org/publications/index.htm

Including chapters 1 and 8-10 of his

*Dynamical Systems in Neuroscience*.

http://www.izhikevich.org/publications/dsn.pdf (PDF!) - notice that he gives the geometry of limit cycles for non-linear systems. This sort of modelling was very popular in the late 1990s Neural Net craze, and lead to 'pulsed neural network' (PNN) models. Applications to neurobiology, bioinformations, chemistry, statistical mechanics, and doubtless to finance (see two state Markov models and 'heteroscedasticity' as terms).

Non-linear dynamical equilibrium is a very general phenomenon, and one philosophical observation I would make is that Jaap Bax's 'dynamical laws as essence' has a direct connection to Izhikevich's wonderful elaboration. Specifically Izhikevich notes that the dynamical law for neurons has *no relation*, really, to the underlying matter. Like Aristotle, he says that the material has to be *suitable* to receive the (dynamical law essential) form, but that within some constraints, the two are unrelated. Thus, I would conclude, the same matter may be *transformed* into a different dynamical form, and also dynamical forms may be reproduced -- transmigrate, as Plato would say -- in different material substrate.

What is important is the *geometry* of the system, which leads to only

**Four**behavioural types of non linear behaviour. (Not unlike the old theory of humours....).

On a practical note, I will add the original observation, on my part, that data centre performance analysis can be rephrased in terms of non-linear dynamics. A comparison of Neil Gunther's

*Practical Performance Analysis*(an excellent book, if you have to measure computer performance professionally!). See chapters 10-12, give or take, then compare to Izhikevitch for guidance.