We discussed in class the definition of the English word bifurcation. Here it is:

Concise Oxford Dictionary, 8th Ed., Copyright 1991 Oxford Univ. Press: /bifurcation/n. 1. a. a division into two branches; b. either or both of such branches. 2. the point of such a division.

Please read the excellent intro given here and borrowed from the UTEP ``SOS math'' project. When you have more time, you may want to also go to their general differential equations resource.

As explained in class (and perhaps too briefly in the book, in page 99), here is a quick way to look for bifurcation values of the parameter for the one-dimensional differential equation

 d y dt = fp(y)
where ``p'' is the parameter and fp(y) is continuously differentiable on y and p.
1. Solve this set of two simultaneous algebraic equations:

 fp(y)
 =
 0
 d fp dy (y)
 =
 0
for values p0 and y0. Let us call the solutions (p0,y0) the ``critical pairs'' of parameter values and dependent variables.
2. Now, for each critical pair (p0,y0), we look at the parameter p0, and determine if it is indeed a bifurcation value.

This procedure is very similar to the procedure which you followed in calc 1, when, in order to maximize a function, you first found critical points, and then studied, for each critical point, if it was a maximization point or not.

As explained in class, the justification for following this procedure works is that when the derivative in 1. is not zero, one has, for any parameter near p0, some equilibrium near y0 and of the same type (source or sink) and thus there is no change of behavior at that equilibrium point, for the given parameter values. So there is no need to look at the values where this derivative is not zero, and it is enough to look for those for which the derivative is zero. The first equation, fp(y) = 0, of course just says that y is an equilibrium.

Let us work out with this method the example of populations under constant fishing given in pages 101-103 of the book. We have the equation

 d P dt = kP æç è 1- P N ö÷ ø - C
and consider C as the parameter of interest. (Yes , one could also think of k and N as parameters. We are just studying one parameter in order to keep things simple.)

We need to solve the two equations in part 1. above, with y being ``P'' and p being ``C''. Actually, let us practice doing this with Maple. We first write the equations, including computing the derivative:

```f:=k*P*(1-(P/N))-C;
fp:=diff(f,P);
```
and we get that the derivative is
 k æç è 1- P N ö÷ ø - kP N
so we next solve:
```solve({f=0,fp=0},{P,C});
```
getting precisely one critical pair:
 ìí î P = 1 2 N,  C = 1 4 k N üý þ
So, is C0 = kN/4 a bifurcation value of the parameter? Let us see.

What are the equilibria for C < C0?

```solve(f,P);
```
gives us:
 kN+sqrt(D) 2k , kN-sqrt(D) 2k
where D = k2N2 - 4kCN. so there are no (real!) equilibrium points if D < 0, i.e. if kN < 4C, i.e., if C is bigger than C0, there is exactly one equilibrium if C = C0, and there are two equilibria if C < C0. This means that yes, C0 is a bifurcation value of the parameter (``the number or the type or equilibria changes'').

You could then go on and decide that types of equilibria we have when C < C0 and at C = C0, as done in the book.

Is this method easier than just doing as in the book? For higher dimensional problems, it does give a good systematic procedure. For the one-dim problems we study here, it depends on the problem. If the equations for the equilibrium are hard to solve, having the second equation may help a lot. To take an example I just made up, say we have

 fp(y) = sin(y) - p cos(y)  .
There are many possible equilibria, for any given value of p. (For example, when p = 0 we have just sin(y), which has zeroes at all the integer multiples of p.) But, if we consider the simultaneous equations:
 siny - p cosy
 =
 0
 cosy + p siny
 =
 0
we can eliminate p to get sin(y) = cos(y) = 0, which has no solutions. So we know that there are no bifurcations , with very little work. (Actually, we could also have noticed that the function fp(y) can be rewritten in terms of a single translated sine, so it would be clear from the graph that the derivative is nonzero at each root.)

Here are a few general references, and also a couple of scientific papers (in web readable form) which describe bifurcation behaviors in some applications:

• lecture notes on chaos and bifurcations (more for discrete systems, but same idea). Has very nice pictures.

• a page on chaos

• Chaos theory: A brief introduction

• Chaos Demonstrations from Caltech

• Chaos and fractals web site

• The chaos game by Devaney

• more Chaos notes

• Here is a picture of the bifurcation behavior for a two-parameter fishing model. The narrative accompanying it is as follows:

A population model for haddock (Melanogrammus aeglefinus L.) developed by Horwood (Phil. Trans R. Soc. Lond B 350, 1995) is analysed further with respect to its ecological stability. It is shown that the dynamic properties are influenced primarily by zooplankton production and harvesting intensity. The derived results relating to ecological stability are compared with available information from the North Sea and the Georges' Bank ecosystems. For a wide range of realistic parameter values, the predicted dynamics are characterised by fixed point dynamics; then the population is primarily destablised by overfishing. High zooplankton production, caused by either trends or fluctuations in production, may, however, drive the population into a region characterized by periodic fluctuations of varying fixed periods, and even aperiodic dynamics of closed curves and chaos. It is a argued that assumed increased climatic variability may change the stability properties of the ecological system.

• Bifurcations in nonlinear convection by Dr A.M. Rucklidge. Abstract:

The aim of this course is to develop the theory of bifurcations in dissipative nonlinear systems, and to show how how these techniques can be applied to specific physical problems. Convection in a fluid layer heated from below is a classic example of a system in which successive bifurcations lead from a trivial static state through ordered behaviour to disorder; moreover, the theory has important astrophysical and geophysical applications.

• Injection-induced bifurcations of transverse spatio-temporal patterns in semiconductor laser arrays by D. Merbach, O. Hess, H. Herzel, E. Scholl Abstract:

We present results of numerical investigations on the complex spatio-temporal dynamics of semiconductor laser arrays. The diffusion of charge carriers turns out to be essential for instabilities in the output intensity above the laser threshold. Besides other bifurcations, a period doubling of a torus is found. The Karhunen-Loeve decomposition gives the dominant modes of the spatio-temporal dynamics of the output intensity and provides a measure of the number of spatio-temporal degrees of freedom.

• Analysis and modeling of bipedal gait dynamics

The main focus of the present investigation is the development of quantitative measures to assess the dynamic stability of human locomotion... ... accommodates the study of the complex dynamics of human locomotion and differences among various individuals.... Changes in the stability of the biped as a result of bifurcations in the four-dimensional parameter space are investigated.

• bifurcation notes by Devaney (author of our textbook)

• and when you want to laugh, follow this link (sorry - I could not resist it - I got it by Altavista searching for ``bifurcations''; if you understand what this is about, let me know) :-)

File translated from TEX by TTH, version 2.00.
On 13 Feb 1999, 20:12.