CHAOSCHAOS THEORYTHEORY !! What is The Butterfly Effect? In 1800 the philosopher Johann Gottlieb Fichte wrote "you could not remove a single grain of sand from its place without thereby... changing something throughout all parts of the immeasurable whole".
In 1890 the mathematician Henri Poincaré noticed that very small changes in initial conditions can cause major changes in the output of the model for The Three-Body Problem.
In 1952 Ray Bradbury wrote a science fiction story in which the death of one butterfly had a far-reaching ripple effect on later historical events.
It was, however, the meteorologist Edward Lorenz who first made popular the term Butterfly Effect, after the publication of his paper in 1972, with the eye-catching title:
Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas? The flapping wings represent very small changes in the initial conditions of a model, which cause large changes in the outcomes of the model. How did Lorenz discover this Butterfly Effect?
In 1961 he was re-running a model to predict the weather for the next two months, the imput data was on punched tape fed into an early valve-powered computer, with no monitor of course, only a printer, those were the hard days (!). For the 2nd run he entered an approximate value of 0.506 from the first run, in- stead of the full precise value 0.506127.. He was astonished to see the new output was now a completely different weather scenario!
At first the new values repeated the old ones, but soon the differences steadily doubled every four days, until the original predictions disappeared somewhere in the 2nd month! The initial round-off errors steadily led to very different weather predictions! Here is an actual plot used by Lorenz with predicted values of the weather along the y-axis and time along the x-axis. The red line represents old predictions and the blue line new predictions with new initial conditions which were truncated (or cut off) versions of the old ones.
Use estimated temperatures for the days 1-17 to answer these questions: (1) On what day did the lines first start to separate? (2) On what day did they first re-join? (3) What day did they next cross over? (4) Which line had the lower temperature on Day 8? (5) On what day was the temperature difference a maximum? (6) The two lines were quite close until which Day?
What was Lorenz's Model? Lorenz used three differential equations to model or simulate the weather conditions in the atmosphere, which was all the time being cooled from above and heated from below.
It became clear to him in 1969 that no matter how accurate the measurements were, the weather could NOT be predicted with certainty because of the Butterfly Effect. Weather patterns such as lightning and tornados are chaotic in nature and some are also fractal in appearance!
When Lorenz used slightly different initial conditions the values they produced behaved chaotically. Each run of the model never met or crossed another run, and together they formed a beautiful fractal which Lorenz called a strange attractor. Notice also that this image shows self-similarity.
Biological Population Dynamics of the micro-organism Yeast The graph opposite shows how the population density (on the y axis) of a micro -organism changes in time (on the x axis).
Iteration models can analyse what happens here, and in other animal and bacterial populations, so that we can help populations to survive and not become extinct.
First write answers to these questions: (7) Write out in full decimal notation 106 and 104 Can you see why we prefer the shorter power notation? (8) Explain what bistable means. Explain why we might say from the graph that the population is actually tristable. (9) Explain what has happened to the population for the red lines at the bottom of the graph. (10) What do you think will happen after 8, 10, 100 days according to this graph? Can we really predict so far? (11) Explain the difference between population size and population density .
In the last Fractal & Chaos Theme using our iterative model you discovered for 0 < b < 2.9 the population of kangaroos on Kangaroo Island converged to a STABLE state of one attractor, either zero or a positive number (like the Yeast in the above graph). So now we need to use your same model to find out what happens for b > 2.9, Do we find stable or unstable states (or both) for the above Yeast population? Back to Kangaroo Island... and the Logistic Equation
You may not have realised after all your work on Kangaroo Island that you have actually met the Butterfly Effect already! You investigated how small changes in b in the formula below could lead to period doubling, then four attractors, then 8,16,...128.... and finally to chaos itself! Just like Lorenz, you found that our model of population growth could become unstable and chaotic when you made small changes in the initial conditions (the value of b). This is the Butterfly Effect! The iterative model we used before was x b*x*(1000 - x)/1000 Check the right hand side is the same as b*x* ( 1 - x ) 1000 Notice that x is the fraction or percentage of kangaroos on the island. 1000 If we now use x NOT as the number of kangaroos but the decimal fraction of the kangaroo population (i.e. the population density) so that 0< x <1 then the model we used simplifies to: x bx(1-x) For example x=0.6 means 60% of the population and so on. We do not NEED to know the actual population when we know the population density, notice this appeared above in the graph about the micro-organism Yeast. This simpler form of the iteration is called the Logistic Equation which is a model widely used to study the growth of animal and bacterial populations. You will now create spreadsheets and graphs from the Logistic Equation to investigate what happens to the Yeast population: to survive or not to survive? That is the question! Below is some practice just to get used to the NEW values for x in the Logistic Equation formula x bx(1-x)
(12) Write the following percentages as decimal fractions, e.g. 67% = 0.67. 32%, 56.1% (be careful here!), 296%, 0.0003%. (13) Write the following decimal fractions as percentages, e.g. 0.04 = 4% 0.99, 0.78, 0.7789, 0.0005, 0.834 (14) Use a calculator to to find the reaults when you substitute x and b in the logistic equation bx(1-x): b = 3,1, x = 0.657, b = 3.111, x = 0.88, b = 0.767, x = 0.967 (15) Explain precisely what is wrong with each of these values of b and x: b = 6.1, b = -4, x = 23.562, x = -0.786 We will now try values of b from 2-4 to see what happens to the Yeast population x after many iterations. If you have problems with the spreadsheet below study the Introduction to Excel help sheets at the end of this theme.
Create a new Spreadsheet 7 exactly like the one below. Put the starting population x = 0.6 in cell I1 and copy it into all the cells A5 to L5.
Copy the values for b (from 2 - 4) you see entered in cells B3-L3
In cell B6 insert the formula = B$3*B5(1-B5)
The $ sign forces the 3 to stay rather than changing as the formula moves down the column. So in this column the b value of 2 remains the same throughout column B.
Now fill across from B6 to L6. Then fill down columns B6-L6. You will need to go down a long way to see the full behaviour of ssome of the populations!
Save and Store this as Spreadsheet 7
Here is the spreadsheet with the first few rows filled in:
What did you discover happens to the Yeast population for various values of b? Here is what should have happened in some cases:
Copy and fill in the table below to include all your results from your Spreadsheet 7.:
Value of b 2.0 2.2 2.4 2.5 2.6 2,8 3.0 3.2 3.4 3.5 3.6 3.8 4.0 Attractors 1 1 2 4 0
Did you find 8 or more attractors? For some values of b you must go down 1000s of rows to see the pattern! (16) List values of b that produce (i) stable and (ii) unstable Yeast populations.
Graphing your results Patterns can often be seen more clearly on a graph rather than looking at reams of numbers. If you can, create your Spreadsheet 8 by copying the spreadsheet below which includes a graph of the results for b = 2.8. More graphs of results showing what is happening to the Yeast population.
Now we see that this is a slow moving graph which will eventually end in two attractors.
This settles down more quickly to give four attractors.
Finally we see eight attractors! Below is one of the most important graphs to help us understand the problem of stable or unstable states for the Yeast population data.
We plot b values along the x-axis from 2 to 4, and along the y-axis the number of attractor(s) in the population for that value of b.
One attractor so far now 2 now 4 now chaos! Can you see what is happening? At first, on the left, there is only one attractor as b increases, until bifurcation (branching) occurs resulting in 2 attractors (period doubling), then 4, 8, 16 and so on leading to chaos.
In the enlargement of the right hand above we see a region of chaos, but notice there are white spaces that occur amongst the chaos. Some of these have 3, 6, 9... attractors that correspond to partially stable values of the Yeast population. even though they are in the midst of chaos! Notice that this image is a fractal because when we enlarge it we see the same shape appearing (self-similarity). In fact if we look at the values in the base line of the Mandelbrot Set we would produce exactly these values. For a very clear explanation of all of this, take some time to look at this video: https://www.youtube.com/watch?v=ETrYE4MdoLQ CChhaaooss TThheeoorryy inin tthhee RReeaall WWoorrldld For such a relatively new field, Chaos Theory has already established itself both in mathematics and in practical applications in the real world, including fluid flow, predicting gas solubility, monitoring heart-beat irregularities, even analysing fluctuations in financial markets! In the late 20th century, Palestinian mechanical engineer Ali H. Nayfeh applied chaos theory in mechanical and engineering systems including the construction and maintenance of many machines and structures that are common in daily life. In celestial mechanics Chaos Theory helped discover chaotic motion in our solar system, for example four of the five moons of Pluto rotate chaotically We will conclude with two of the most significant discoveries related to Chaos Theory that raise a deeper question about the nature of the universe itself!
The Three Body Problem and the Double Pendulum
If we know the initial positions and velocities of three bodies (e.g. planets) can we predict in general what happens to them? This is the Three Body Problem discussed by Galileo Galilei in 1499, Isaac Newton in 1687 and Henri Poincaré in 1890. Even using Newton's EXACT deterministic Laws of Motion and Gravitation it is still not possible to find a general solution because the resulting dynamical system is chaotic.
There is a way of simulating the three body problem here on earth in the comfort of your own home - the double pendulum - two pendulums attached to each other but free to move independently. The three body problem has only two degrees of freedom, relative to one of the bodies there are only two motions to analyse, just like the double pendulum.
Here is our friend Eckhard Müller and his double pendulum attached to the wall. Both red and blue pendulums are free to move. (the image of the pendulum is blurred because it is in motion!)
Some of the positions of this double pendulum in chaotic motion: The internet has many videos about making and using Double Pendulums. Here is one: The same double pendulum is released six times from the same start position and is filmed in all six cases to see what happens. On the right are the different positions of the six pendulums after the same period of time! Very small differences in the start position (initial conditions) for the six runs causes different end results - the Butterfly Effect !
A Simple Diode shows Chaos in Electrical Circuits!
Above is the Chen and Ueta circuit 2002 whose behaviour is chaotic with a complex attractor (above) that reminds us of the Lorenz strange attractor.
How is this possible when the laws governing electrical circuits and systems are deterministic? It is exactly the same dilemma that we saw with models for weather prediction, biological population growth, and the three body problem! The most remarkable thing about all of the above examples is that they arise from completely different and seemingly unconnected fields: meteorology, biology, physics and electrical circuits, all have deterministic laws and yet they all exhibit chaos! To Conclude
In 1975 the physicist Mitchell Feigenbaum Most Important Maths Numbers discovered there was a universal number that occurred in ALL models of real life situations that showed period doubling and chaos. This was called the Feigenbaum Constant which is number 5 on the list of the most important numbers in mathematics (right):
The number is not known exactly but can be calculated to any degree of accuracy and is: 4.669201609......
For a long time it has been believed that our universe worked rather like clockwork, where deterministic scientific laws could precisely predict the future.
Chaos Theory, however, shows us that this is not true, and that the physical world, and life itself, will always be unpredictable and chaotic at times!
So maybe we do live in a universe where fractals and chaos are everywhere? What do you think?