Introduction into chaos

A good starting point for bluffing your way in life is the series of books Bluff your way in ... " We try to give you in this first hour a short

"Bluff your way in chaos,fractals and differential equations".

This first section should be a {\bf motivation} for more serious material, which follows in the next weeks.

Here is the Bluffer's definition of chaos

Chaos is still in fashion. From 1991-1993, 2406 journal articles and in 1993-1995 even 2451 articles use the word chaos in the title or abstract. This means that you had to read in average 6 articles per day during that time in order to stay up to date. And then you have not yet read most of the good articles, where the word chaos does not appear. The chaos blow reached newspapers, TV and even the film, pictures of Mandelbrot and Julia sets are hanging in living rooms or banks, some books got bestsellers. However, in the nineties, the public or at least the journalists have got a bit tired of it: everybody has seen the Mandelbrot set many times on photos, posters, newspapers, TV, school and on home computers. Still it is not possible to use the insights of the "chaologists" to predict for example the Dow-Jones index for the next week. Some newer research has therefore been devoted to the "control of chaos", a tipic, which was now fashion for a few years. It is not an easy task to answer the question

What is chaos?

The problem is that there are simply too many definitions for this term. One has to use finer definitions according to the categorie of dyanamical systems used. Roughly speaking, chaos means "sensitive dependence on initial conditions". I learned this by looking the movie "Jurassic parc". Chaos is usually explained by stories: a famous popular story is the butterfly of Lorentz. The picture is that the motion of a butterfly can produce a perturbance of the athmosphere which results into thunderstorm within several weeks. An other story tells that the gravitational force of a little bug on Sirius can change completely the orbit of a billiard ball after it has hitten enough times the boundary of a suitable table. This statement is even proven to be "true" if the billiard table has the shape of a stadion built by pieces of circles and straight lines. A much more simple experiment which shows the "butterfly effect" in a wonderful way is to program the function $x \mapsto 4 x (1-x)$ in two different ways and to compare the outputs. Let's do that in Mathematica (see also the seperate link)

T[x_]:=4 x *(1-x); IterateTT[x_,n_]:=Module[{y=x},Do[y=T[y],{n}];y]; S[x_]:=4 x- 4 x^2; IterateSS[x_,n_]:=Module[{y=x},Do[y=S[y],{n}];y]; {IterateTT[0.1,59],IterateSS[0.1,59]}; ListPlot[Table[IterateTT[0.1,k]-IterateSS[0.1,k],{k,100}]]

Note that Mathematica computes with 16 digits. This does not help! The results of the two outputs of T^{59}(x) and S^{59}(x) are completely different and illustrate that the computer does not follow the associative law during the computation. The same experiment can also be done on any programmable calculator. Because such instruments calculate with less precision, the effect can be seen even with less iterations. By the way, the map

x --> 4 x (1-x)

was proposed as a random number generator by S.Ulam and J.von Neumann, who had been interested in the design of algorithms for random numbers on the first electronic computer. An other story is the true tale that if you look in the television how the balls of the numbers pool (Lotto) are falling and you shake your hand, it will affect the motion of the balls. The little gravitational change over hundreds or even thousends of miles has already an affect on the balls and will change the numbers and also the winners. Apropos television. If you have a video camara, you can film inside the picture shown by the camara. Strange patterns appear. Chaos does not only have its root in nonlinearity: The cat map

(x,y) ---> (2x+y,x+y)

on the torus is a linear map but a poor cat sitting on the torus and exposed to the dynamics will be in many little pieces after only a few iterations. The idea to put a cat onto the torus is in Arnold's great book in classical mechanics and the origin of the name of the map. In the next experiment, we see just one orbit of length 1000 of this map: By the way, it is a shame to use more than a line for such a program.

T[{x_,y_}]:=N[Mod[{2x+y,x+y},1]]; ListPlot[NestList[T,{0.2,0.4},1000]];

What are fractals?

A fractal is a subset of a metric space with "fractional" Hausdorff dimension . The most famous non fractal set is the

Mandelbrot set : it's Hausdorff dimension is 2! It is not fractal according to the above definition, a fact which however does however not affect the beauty of the pictures. A calculation of the Mandelbrot set can be done by a two line program (I tried hard to get it on one line but I did not yet succeed).

M=Compile[{x,y},Block[{z=x+I y,k=0},While[Abs[z]<2&&k<50,z=z^2+x+I y;++k];k]]; DensityPlot[50-M[x,y],{x,-2.,1.},{y,-1.5,1.5},PlotPoints->200,Mesh->False];

The name Mandelbrot was invented by Douady and Hubbard who investigated this fascinating object the first time mathematically. The first picture of the Mandelbrot set has been computed not by Mandelbrot but by Brooks and Matelski who studied Kleinian groups. Their name is however not forgotten: there is an open subset of the Mandelbrot set which is now called Brooks-Matelsky set . It is a big unsolved question, whether this set coincides with the interior of the Mandelbrot set. Douady and Hubbard have shown that this is true if the Mandelbrot set is locally connected. The later problem is the big yet unsolved question in the field. Beautyful sets are the Julia sets. They live in phase space of the dynamics opposit to the Mandelbrot set which is a subset of the parameter space. Nevertheless, the numerical computation goes very similar with a two line Mathematica program. This is the most famous Julia set, the Douady rabbit:

J=Compile[{u,v},Block[{z=u+I v,k=0},While[Abs[z]<99&&k<50,z=z^2-.12+.74I;++k];k]]; DensityPlot[50-J[u,v],{u,-1,1},{v,-1,1},PlotPoints->200,Mesh->False];

Fractals are known over 100 years now. The most famous and ancestor of all fractals is the Cantor set, which is the prototype of a fractal. Did you know that it has not been found by Georg Cantor but by Henry Smith in 1875?

What is the relation of fractals with dynamical systems?

There are many relations:

Attractors or repellors of dynamical systems are often fractals. An example are Julia sets.
Subsets of parameter planes look often fractal. An example is the Mandelbrot set.
Dynamical systems can create self-similar fractals. An example are iterated function systems. Moreover, there is a natural chaotic map associated to each attractor of an iterated function system.
The closure of an orbit of a dynamical system has a fractal structure. An example is the Standard map. Here is an orbit of the standard map.

T[{x_,y_}]:=N[Mod[{2x-y+1.3*Sin[x],x},N[2Pi]]];ListPlot[NestList[T,{.2,.4},10^4]];

When speaking from fractal invariant sets of dynamical systems, it is good to know that the famous "strange attractors" are only conjectured. Mathematically, it is in general difficult to estimate the dimension of an attractor from a specific case like the famous Henon map, which we see next:

T[{x_,y_}]:={0.3*y+1-1.4*x^2,x};ListPlot[NestList[T,{.56,.37},10^4]]

For this choice of parameters, it is still not known (and maybe not a good question) whether we see a "strange attractor". A little bit gossip also here: the publication of the first papers about strange attractors had some difficulties in acceptance. Henon's paper was first not accepted because it was believed that the attractor is just a periodic orbit of large period. Also Ruelle had problems with the publication of the Ruelle-Takens paper. He had to be his own referee in order to accept the paper with Takens. Also Feigenbaum had problems to publish his results about the Feigenbaum attractor. They are indeed hard to read. The question

What is turbulence?

is believed to be one of the big open problems of our time. One difficulty is that even the definition of turbulence is not so clear. One could say that turbulence is chaos for partial differentical equations. Some people believe that "space-time chaos" is a good working definition for turbulence. Space time chaos means that not only the time evolution is chaotic but also space translation. Think of drop of blue ink falling in water. Even if time is stopped after a while, the shape of the blue color is after some time very complicated. If two space coordinates are considered as space and the third as time we have something like a chaotic dynamical system.

Is dynamics useful?

The easiest reply to this question is: never ask a mathematician about usefulness, mathematics is art! But since it is usually not the mathematician who decide how much money goes into this science, one has to reply more carefully and argue that chaos might help to understand the weather, the stock market, the health of the brain and the heart. Let us concentrate on the weather prediction: already at the beginning of this century, Poincare realised, that the long time predictions of deterministic systems are problematic because systems can be chaotic and discussed already the problem of making weather predictions. The ideas were forgotten for a while and when they were rediscovered, when computers were available. An early computer study was the analysis of the Lorenz system in 1963.

Orb=NDSolve[{x'[t]==10(y[t]-x[t]),y'[t]==-x[t] z[t]+28x[t]-y[t],z'[t]== x[t]*y[t]-8z[t]/3,x[0]==z[0]==0,y[0]==.3},{x,y,z},{t,0,40},MaxSteps->5000]; Plo=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]} /. Orb],{t,0,40},PlotPoints->5000]; Do[Show[Plo,ViewPoint->{2k,1,1}],{k,-2,2}]

This is a simple toy model for the time evolution of a convecting fluid layer. Other very complex system are the stock market or neural networks like our brain. We don't know even the exact laws for such complex systems but we can measure datas like temperature and pressure for the climate, the Down Jones index in the economic world or the electroencephalograms for the brain. It would be very interesting (and profitable) to know, if such complicated dynamical systems could be simulated by low dimensional dynamical systems. Recent research has suggested that chaos is normal in the heartbeat and contrary to expectations, it is the sickest hearts whose beat often looks most periodic. Similarly, levels of white blood cells fluctuate chaotically in healthy people but oscillate cyclically in certain patients with leukemia. There are also indications that the immune system's method for making antibodies may involve chaotic activity. Tremors of Parkinson's disease may arise from a loss of normal chaos in the neurological system. To answer the question about the usefulness of chaos: there is no doubt that the ideas of chaos had a tremendous stimulation for a part of science and even if direct applications are not possible, the ideas have increased the knowledge about many things.

In order to bluff your way more successfuly, we recommend to look at some of the following books about chaos. It is no bluff however, that there are many more on the market. You find a literature list with books I recommend if one or the other of you might want to know more during the course or later. The theory of chaos and fractals are popular pillars of the theory of dynamical systems. Lets ask our last question:

What is dynamics?

According to a mathematician, a dynamical system is a "semigroup acting on a set". Less bluffy: dynamics , the theory of dynamical systems deals with maps or differential equations. The aim is to understand its qualitative and quantitative behaviour in time. We will in this course deal both with maps and with differential equations.

Summary

Dynamics is the theory describing the iteration of maps or evolution of differential equations.

is sensitive dependence on initial conditions of dynamical systems. We illustrated that in class by showing that it is not possible to determine with a computer the long time behaviour map in general. {\it Fractals} are sets with noninteger Hausdorff dimension. Invariant sets of chaotic dynamical systems are often fractal. On many fractals there is defined in a natural way a chaotic dynamical system.

Turbulence} is space-time chaotic behaviour of infinite dimensional dynamical systems like partial differential equations.

A final word

Even if there are doubts that chaos theory is useful to understand completely complex dynamical systems in economics, medicine, atmospheric sciences, it stimulated the scientific work of many people and leaded to progress which would probably not have been achieved without this change of view.