by Colin Pask
Bob May said we should experiment, so here we go. Suppose we take λ = 3.2 for which xeq = 0.6875. As an experiment, take an initial population x0 = 0.6. Using equation (12.14), we generate the following sequence of population values (rounded to four decimal places):
0.6, 0.768, 0.5701, 0.7842, 0.5415, 0.7945, 0.5225, 0.7984, 0.5151, 0.7993,…
Instead of converging to the steady-state xeq = 0.6875, the population is becoming focused on two populations:
xeq1 = 0.7994555 and xeq2 = 0.5130445.
But now the population does not tend to a single value; it jumps between these two special values. If we had started with a different initial population x0, the result would still be the same; there are now these two special population values, and the evolving population eventually settles down to oscillate between them.
For λ < 3, the population settles down to have the same value at every step, and we may say that it has a period of one. For λ > 3 we find that the population repeats every second step, and this is called period-two behavior. The system is said to have “period doubled.”
Population growth and resource pressures are no longer balanced for these larger growth rates, and there is now a boom-and-bust cycle. The population behavior is a little more elaborate than it was but it still makes sense in terms of overshooting and undershooting.
(A technical aside: imposing the condition that equation (12.14) gives the same result every second time leads to a formula for the two special populations. Let p = ½(1 + 1/λ) then xeq1 and xeq2 are given by p ± √p(2 – 3p). Putting λ = 3.2 will give you the two numbers quoted above.)
12.5.4 Even More Surprises
Increasing the growth constant λ just beyond 3.5 produces another surprise. Now it is the two special population values that lose their stability, and instead we find there are four new special values, and the population cycles through them. Thus it repeats every four steps, and we have a period-four situation. The system is said to have period doubled again. The population booms and busts are now getting really complex!
Perhaps you will guess that increasing λ a little more leads to another period doubling and period-eight behavior. Further period doublings occur as λ increases. It is a bizarre story.
There is one final twist; increasing λ beyond around 3.57 leads to a situation where there are effectively an infinite number of periods, and the output from equation (12.14) leaps around with no discernible pattern at all. This has become known as chaos. It is important to note two points. First, putting an x0 into equation (12.14) will allow us to steadily generate a string of numbers. If anyone repeats that calculation with exactly the same x0, they will get exactly the same string of numbers. It might be said that these are predictable, or determined, outcomes, but to all intents and purposes they look as though they were generated by some sort of random process.
Second, if the input x0 is changed only a tiny amount, the generated sequence of numbers will change and soon become nothing like the original sequence. This is called sensitivity to initial conditions. The simple logistic equation has taken us into a weird world.
12.5.5 Discussion
For a long time it was believed that the world obeyed deterministic laws; the pendulum continues to swing in a regular way, and Newton's mathematical description of the solar system led people to think of a clock-work universe. Of course, there were random effects (wind gusts might disturb a projectile on its regular flight path), but they could be added on, and “noise” was to be expected in any real physical system. There had to be averaging to deal with the enormous number of particles in gases and solids, but still, the deterministic equations provided the starting basis for all of that. It had been a shock to find that at the very smallest levels there was a need to use quantum mechanics and its accompanying probability interpretations. Now came the shock that even the simplest deterministic equation related to macroscopic phenomena might produce a chaotic output.
At this point I need to deal with an obvious question: Is all this talk of chaos just based on that one simple equation? The logistic equation is easy to use and allows its output to be understood in mathematical terms (see May's review). It is the simplest nonlinear difference equation. However, it is now established that there is nothing singular or special about the logistic equation, and period doubling and chaotic behavior have been identified in a multitude of cases. Furthermore, the same sort of behavior has been found for continuously changing systems (see the review by Motter and Campbell for an introduction to the early work of Edward Lorenz and others). The theory has been found to apply in real physical situations.
At this point I should offer a few suggestions for further reading (see bibliography for more information). Collections of articles are given in the books edited by Hall (very readable and as published in New Scientist) and Cvitanovic (seminal papers and reviews—including the work of Mitchell Feigenbaum showing the generality of the chaos picture and how order may be found in the route to chaos). Good popular books are by Gleick (giving the story of chaos research and the people involved) and Stewart (with a smooth introduction to the relevant mathematical ideas). The book by Moon is a good example of the impact of the discovery of chaos on applied scientists and engineers. The review, “Chaos at Fifty,” by Motter and Campbell is a fine introduction to chaos and its discovery in continuous systems. Chapter 11 in the book by Barger and Olsson is a good introduction to nonlinear mechanics and chaos.
The celebrated French scientist and mathematician Pierre-Simon Laplace (1749–1827) was a champion of Newton's methods and showed how refining the use of his gravitational theory could account for many intricate, tiny, and subtle effects observed in planetary motions. His expression of the power of deterministic systems has become famous:
Given for one instant a mind that could comprehend all the forces by which nature is animated and the respective situation of the beings that comprise it—a mind sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.10
We now know that chaos makes a mockery of Laplace's claim, and signs of chaos are beginning to be widely detected. Few could doubt the impact of calculation 50, simply chaos.
The literature on and around chaos is now voluminous and diverse. Chaos theory features in Tom Stoppard's play “Arcadia” and a good conclusion for this section might be to quote from the play's program notes written by none other than Professor Robert May:
There is a flip side to the chaos coin. Previously, if we saw complicated, irregular or fluctuating behavior—weather patterns, marginal rates of Treasury Bonds, color patterns of animals or shapes of leaves—we assumed the underlying causes were complicated. Now we realize that extraordinarily complex behavior can be generated by the simplest of rules. It seems likely to me that much complexity and apparent irregularity seen in nature, from the development and behavior of individual creatures to the structure of ecosystems, derives from simple—but chaotic—rules. (But, of course, a lot of what we see around us is very complicated because it is intrinsically so.)
I believe all this adds up to one of the real revolutions in the way we think about the world.11
12.6 APOLOGIES
I must apologize to those readers who work in the many areas of modern science that are not represented in my list of calculations. This chapter has hinted at the importance of probability theory in science and, for example, the studies of phase transitions using the Ising model and other parts of statistical mechanics are sadly missing. There has also been a great increase in methods of data collection and its statistical analysis, which might have been included. But fifty calculations I said, and so just fifty it remains.
in which I suggest how the calculations may be viewed in various contexts, explain how I came to choose them, and pick out ten of them to be labeled “great.”
; Here is a list of the fifty calculations described in the preceding chapters. The number after the calculation name gives the section in which it is described. The letters and stars refer to attributes of the calculations that I will explain in section 13.2.
1. Malthus on population growth
2. Mesopotamian Pythagorean triples
3. Archimedes bounds π
4. Fibonacci's presentation in Liber Abbaci
5. production of tables of logarithms
6. Euler solves the Basel problem
7. the prime number theorem
8. Eratosthenes measures the earth
9. Kelvin and the age of the earth
10. seismic rays reveal the earth's interior
11. Galileo describes projectile motion
12. tide predictions
13. Ptolemy's Almagest
14. Kepler's astronomical calculations
15. Newton's Moon Test
16. Newton's determination of planetary masses
17. predicting the return of Halley's Comet
18. the discovery of Neptune
19. finding the astronomical unit
20. rotating orbits
21. why the night sky is dark
22. state of the universe
23. Hoyle makes carbon
24. galaxy rotation and dark matter
25. escaping gravity
26. Harvey establishes blood circulation
27. Halley values annuities
28. the Hardy-Weinberg law
29. the mathematics behind the CT scan
30. scaling from mice to elephants
31. light has a finite speed
32. seeing a rainbow
33. diffraction and the limit to vision
34. light and electromagnetism
35. photons exist
36. bending light
37. atoms really do exist
38. spectral line patterns
39. the new mechanics explain atoms
40. discovering the mass of the neutron
41. the electron magnetic moment
42. why we must have a neutrino
43. decays and time dilation
44. quarks tell us particle masses
45. why the sun shines
46. planning for a bomb
47. strings and Fourier's mathematical poem
48. tabulating Bessel functions
49. FPU and oscillation surprises
50. simply chaos
13.1 LEARNING FROM THE CALCULATIONS
The calculations are part of the story of science and mathematics and provide a selection of some of the key advances in these areas. It is possible to take these calculations as a little database that may be used in different ways to extract information about certain topics. I will briefly introduce three examples. In the following discussions, I refer to the relevant calculations by giving their number, for example [12] indicates calculation 12, tide predictions.
13.1.1 The Progress of Science
The calculations described above give us a historical tour through physics.
The early calculations show how astronomers like Ptolemy, Copernicus, and Kepler [13, 14] used data to fit a mathematical model of the solar system in order to generate tables for observers to use. Kepler [14] went further, and from his calculations he deduced Kepler's laws, which summarize the dynamical behavior of the planets. Boyle did a similar thing for gases, giving us Boyle's law linking the pressure and volume of a gas.
A new era began when Newton presented his theory of classical mechanics, and then calculations [15–20] were required to check that it accurately accounted for the solar system. It was also used in continuum systems and vibrations of strings and membranes were studied [47]. Eventually, mechanics also led to kinetic theory and an explanation of Boyle's law [37]. The concepts of mechanics could also be used to describe and explore properties of the earth, its surface phenomena [8, 11, and 12], its internal constitution [10], and even its age [9].
It has always been a struggle to find suitable theories for light and explain optical phenomena [31–33, 35]. Classical physics was completed by Maxwell adding his electromagnetic theory to Newton's mechanics and then producing an electromagnetic theory of light [34]. The fundamentals of science were extended with the arrival of the theory of relativity [22, 25, 36, and 43] and the validating of the atomic hypothesis [37]. Calculations showed how atoms and light were connected [38, 39].
Gradually, the building blocks of matter, or the fundamental particles, were discovered and their properties established [35, 40–44]. Nuclear physics finally explained some astrophysical mysteries [23, 45] and, sadly, gave us terrible new weapons of war [46].
Finally, a new way to do science using mathematical experiments became feasible with the advent of the electronic computer. Calculations using computer simulations [49, 50] have taken us back to classical mechanics and shown that there were still strange things to be discovered.
Throughout all of the work based on and around these calculations, there is the central idea that the theoretical results must match the experimental data for the work to be considered as real science. Some modern theories, like “string theory,” have yet to produce any testable theory-experiment links and so (quite rightly in my opinion) many people do not accept them as genuine parts of science.
13.1.2 Mathematical Methods
The first calculations usually involved a combination of geometrical ideas and simple arithmetic. Eratosthenes's calculation [8] of the radius of the earth is a good example; it also emphasizes the point that only a simple calculation may be required to establish a result of great significance. Over a thousand years later, the use of the transit of Venus to measure the size of the solar system [19] still relied on that geometry-arithmetic combination, although now the details are rather more intricate. Similarly, the modeling of the solar system by Ptolemy [13] and Kepler [14] was based on Euclidean geometry.
Galileo used geometry to represent physical quantities when he constructed what today we call the speed-versus-time diagram [11], so that a geometrical area now measured distance traveled. He too combined arithmetical arguments with his geometrical approach to produce a theory of projectile motion and range data.
The major change came in the seventeenth century when Newton and Leibniz introduced the calculus. Gradually, analytical methods became dominant, although it must be remembered that Newton's approach in the Principia is still very much geometrical, and his Moon Test calculation [15] is still in the old geometry-plus-arithmetic tradition. The work of people like Pierre Varignon, Lagrange, Euler, and Laplace created the analytical form of classical mechanics that has come down to us today. Thus the equations of motion were formulated, and the task was to integrate them in particular cases. The prediction of the return of Halley's Comet [17] is a good example of how these new approaches still required great numerical work. The discovery of Neptune [18] was another impressive example, and, in recent times, there have been wonderful calculations of rocket trajectories and spacecraft orbits [25].
While it is still common practice to plot experimental results on a diagram, a new mathematical task also arose: find a formula that fits this data and reveals the key parameters controlling it. A whole range of mathematical and statistical techniques had to be developed. Examples in this area are Lord Kelvin's work on tides [12], the scaling properties of animals [30], Balmer's discovery of order in the hydrogen spectrum [38], and Hubble's study of the spread of galaxies [22].
The use of mathematics to interpret physical data forced researchers to confront inverse problems, which are often mathematically difficult and require sensitive handling of the input data. Thus the discovery of Neptune [18] required the deviations in the orbit of Uranus to be used to predict the existence of planet Neptune, whereas in the direct problem, we would take Neptune as a known planet and calculate how it affected the orbit of Uranus. Other inverse problems arose in seismology [10] and form the basis of CT scans [29], whi
ch have revolutionized parts of modern medical science.
The change from geometrical methods to analytical approaches also allowed scientists to manipulate theories and combine different concepts and known results. Perhaps the best example is the way Maxwell combined electric and magnetic phenomena to produce his equations as the basis for electromagnetism [34]. Combining the conservation laws for energy and momentum led to an analysis of beta decay [42], which ultimately led to the discovery of the neutrino. Einstein was able to manipulate his theory of general relativity to show the way in which light paths must bend when they pass near a massive object [36].
To explore theories and fit them to physical data required great skill and effort. A new approach involved analogue devices. Thus Thomas Young explored wave theory using a ripple tank [33], and Lord Kelvin invented a machine for calculating tides [12]. Lord Kelvin (as Sir William Thomson) and Peter Guthrie Tait included in their Principles of Mechanics and Dynamics a whole appendix on “Continuous Calculating Machines.”
Gradually, the availability of mechanical calculating machines increased, and much of the arithmetical drudgery was eliminated. (See the book by Michael Williams for the full story.) However, the whole nature of the scientific enterprise was changed with the arrival of the electronic computer. Now it was possible to evaluate formulas, fit formulas to data, and even do algebra, so that enormous calculations—like the calculation of particle masses using quark theories [44]—could be carried out without the hours or even months of work previously required for such tasks. The new approach, known as computer simulations, also entered science [49 and 50].
Looking back over the calculations, it becomes apparent that tables have played several important roles. There were tables that recorded mathematical results for other calculators to use: Mesopotamian tables on clay tablets [2], arithmetical tables in the Liber Abbaci [4], tables of logarithms [5], and tables factorizing numbers for people like Gauss to use when studying prime numbers [7]. Tables recording population details were important and used by people like Halley to calculate things like annuity values [27]. Astronomical tables have always been important from ancient times and on to Ptolemy [13] and Kepler [14]. Warfare required ballistic tables, and Galileo showed how to calculate them on the basis of his projectile theory [11]. Tables have been of great practical importance; for example, Lord Kelvin showed how to calculate them for tides [12]. As analytical methods began to dominate theories, it was necessary to produce tables of special functions, and those for Bessel functions are an outstanding example [48]. It is easy in this age of the electronic computer to forget the parts played by tables in almost every area of science. (The history of table-making is comprehensively covered in the book edited by Campbell-Kelly and colleagues.)