As a relatively new and exciting science, chaos science grew very slowly during its infancy. Yet in the last decade, due to active research in many areas, it has became one of the hottest topics in academia as well as the popular science press. Many books have been published, and millions of Internet pages have been designed full of fractal pictures.1 Given that chaos is associated with disorder or confusion in a system or condition, why does it continue to attract so many people?

History of chaos

To understand chaos, one first has to understand the Newtonian worldview. Sir Isaac Newton’s (1642-1727) development of the calculus and laws of classical mechanics began a scientific revolution in seventeenth-century Europe that caused all subsequent scientists to view nature from a profoundly different perspective. Now that they finally could determine the dynamics of bodies by simple equations, they believed that they had found the ultimate eternal rules that shape the universe.

French physicist Pierre-Simon Laplace (1749-1827), who based his work upon Newton’s work, is credited with the following famous quotation (often referred to as Laplace’s Demon): ‘We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.’2

Laplace’s Demon states the idea of determinism, that the past completely determines the future. One can clearly see why determinism was so attractive to scientists at that time. However, in Laplace’s word everything was predetermined: no chance, no choice, no uncertainty. A solid, inevitable destiny had frozen the events in every corner of the past and is continuously spreading out to the future to do same there. Determinism apparently invokes the idea that whole universe is like a clock. God set it in motion at the beginning of creation and then removed Himself, for everything had been planed before. The ideas that there was no place for free will and that God could not interfere killed the belief in a soul and, consequently, in spirituality. Philosophers and scientists have discussed this for many years. Determinism affected many philosophies and triggered the major ideological movements of during eighteenth century, especially in Europe.

Toward the end of the 1800s, mathematicians and scientists began encountering some very difficult equations, some of which we know today are unsolvable. The most troublesome are various nonlinear differential equations. Even though it looks like such a simple and totally deterministic system, the problem of three bodies attracting each other with purely gravitational forces (e.g., the sun, Earth, and moon triple) turns out to be missing an exact solution. At first, such problems were cast-off as special cases and largely ignored.

One reason for this also might come from the fascinating world of quantum mechanics, which dazzled even the great physicists, and the lack of fast computers at that time. When these equations finally were studied in detail, a fundamental change that would ultimately overthrow determinism began to occur in mathematics and science. An indication of the science that would be come to be known as ‘chaos’ began to appear.

Why does chaos interest people?

In contrast to its common usage, chaos does not actually mean disorder or confusion. By definition, it should occur in well-defined orderly systems. However, most natural physical systems often can exhibit an unpredictable or intractable behavior in the long run, even though the system is defined by clear-cut orderly mechanisms. In this sense, chaos can be defined as unpredictability rather than disorder.

For example, meteorologists use 12 sets of well-defined equations to forecast the weather. They relate such atmospheric parameters as pressure, temperature, flow speed, and time to each other. One can make a computer program that calculates the parameters’ final values by taking any initial conditions as the run’s starting point. In principle, therefore, if we know the initial temperature, pressure, and time values that describe today’s weather conditions, it is possible to derive tomorrow’s weather conditions by running a computer program, which is nothing more than a chain reaction of numerical iterations.

However, in practice, initial conditions cannot be measured exactly and so contain a degree of uncertainty. But since the system’s governing laws are known, one may estimate the effect of errors on future results. Hence, instead of giving the exact results, one can provide an approximate range of possibilities. This range can still be very useful, provided that the deviations do not stray too far from the actual values. In addition, knowing how the error grows in the system might help us understand and control the systems. But if we apply these error estimates to weather forecast equations, we will encounter a large problem, for errors grow exponentially in such systems. Even a tiny deviation at the beginning can create huge deviations from the actual values. It also can provide unrelated or nonsensical results.

An example of chaotic systems

This numerical behavior was first observed by the meteorologist Edward Lorenz, a pioneer in modern chaos work. Fascinated by the results he obtained, in the early 1960s he gave an interesting metaphor to explain the situation of high sensitivity to initial conditions: A butterfly’s slight wing movement (i.e., a little deviation from the initial conditions) can change the future in a way that causes some chain reaction that ultimately result in a hurricane.

Such systems that exhibit a very high sensitivity to initial conditions are called chaotic systems. Chaos comes from the mathematical properties hidden in the equations defining the system, and such unpredictability cannot be removed. Even if the measurements’ quality could be improved by minimizing errors, chaos never disappears from a chaotic system.

One may suppose that chaos occurs in complicated systems, such as weather forecast systems having 12 sets of equations. But even much simpler systems, such as billiards, can exhibit a very high degree chaos. A usual billiard system consists of many balls and a rectangular shaped table. I challenge master billiard players by requesting them to play the game in a stadium-shaped table. I am sure that they will find it difficult to do so, because such billiard tables would be chaotic systems.

If a system is defined as chaotic, this does not necessarily mean that its behavior is totally undefined all the time. As in stadium billiards, a ball has to be inside the billiards, so it should be somewhere on the table, even though sometimes we cannot foretell its exact position because of chaos. Besides, if you send the ball with a velocity perpendicular to a straight side, it will bounce back and forth between the two sides forever. Therefore, depending on which initial conditions are taken, chaotic systems also can show characteristics of regular motion.

A 3-body problem (in general n-body problems) such as the sun, Earth, and moon system, is a chaotic system. But since we can predict the motions of these celestial objects with great precision for many years in the future, why do we call this system chaotic? This triple system possesses a very special set of conditions: distance between bodies, their masses, and their velocities. These parameters cause it to exhibit near-regular behavior. It is analogous to the example of stadium billiards given above, for this triple system bounces back and forth between the table’s sides.

Conditions for chaos

Chaos also can be caused by other factors than just uncertainties measured in the initial conditions. Scientists generally define hypothetical systems by isolating them from the outside world in order to simplify them as much as possible. However, as even objects in the real world that are far apart interact with each other, no system in the real world can be isolated. Given this, a closed (isolated) system might be defined as a fluctuating approximation to its real counterpart, which is changing in an unpredictable manner all the time. In short, even though we would know the exact initial conditions, the actual system could be chaotic due to changes in the approximate system. In this sense, many physical systems have an inclination toward being chaotic.

Due to its maximum complexity, the universe is the largest chaotic system. Observable regular patterns in special parts of that system repeat themselves in time. While the rest of the flows are unpredictable, they are not totally irregular, abrupt, or disordered. Just like whirls in a flowing river, they are in a kind of free motion searching for convenient conditions in which to give birth to organized structures.

The future of chaos

Chaos gives today’s scientist a new worldview, for Newton’s concrete, cold, and deterministic one has been shaken by the uncertainty principle of quantum mechanics. No one ever thought that the Newtonian worldview could be replaced. However, now scientists are more likely to be open to chance and choice than their predecessors. The question is whether chaos theory will cause large revolutions in how we understand the universe. However, the existing excitement, expanding research and growing number of articles, and its numerous applications from economy to biology, seem to indicate that a surprise improvement might not be so far off. ‘

Footnotes

1. Fractal: A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
2. ‘Chaos and Fractals: Laplace’s Demon.’ Online at: www.pha.jhu.edu/ ldb/seminar/laplace.html. 

You would indeed have a lot to calculate before taking off: What is the most economical flight speed of your aircraft, and how does it vary with winds, flight altitude, and the extra weight of fuel reserves? Perhaps the gliding performance of your aircraft is good enough to permit you to travel some distances by soaring in up- draughts, with the engine off. It may even be economical to make detours in order to stay over regions with strong up-draughts. Would it be favourable to bring along large fuel reserves and fly non-stop for long distances without refuelling, or to travel by numerous short flights, saving transport costs for extra fuel? To solve that problem, you have to know about possible refuelling stations along your route, about petrol prices, landing fees, and the time delay you will face by landing at a particular site.

Which is the optimal route to your destination? Perhaps it pays to follow the Great Circle, or should you make a detour where you can benefit from favourable winds? Of course, you will fly only on days when the weather is as favourable as possible, but which is the best weather? And how can you know which weather condition is most favourable for you? What flight altitude will you prefer under different weather conditions? You must also consider whether and how you will compensate for wind drift, when to fly low along coast-lines and other leading-lines to gain protection from the wind and avoid drift, and how to exploit the winds at high altitudes. And you must also take into account many unpredictable situations that you will face. There are lots of questions and you would spend a long time looking up facts and making calculations before you could arrive at a reasonable strategy for your flight.

Your situation is analogous to that of migrating birds. The migratory strategy of birds, a harmonious mixture of rigid and flexible behaviour to achieve a safe and economical journey which can be affected by a bewildering number of factors, is an astonishing feat – where do they get the skills and capabilities to accomplish it?

There are many animals, not only birds, which migrate on a regular basis. Some examples: Bull elephant seals travel 21,000 km (over 12,000 miles) each year, from California to the Gulf of Alaska, the longest migration of any mammal. Millions of monarch butterflies fly 2,000-3,000 km from Canada to the Gulf of Mexico. Grey whales swim 10,000 km from the Arctic Ocean to California and Mexico. In Bracken Cave in Texas 20 million free-tailed bats assemble every summer. All are female. They leave their mates 1500 km to the south in Mexico to come here to give birth to their young. Herring migrate annually and cover over 3,000 km. The bison, the wildebeest, the sockeye salmon, the eel are also known to migrate long distances. But here we will consider the migration of birds – one of the most impressive feats in animal behaviour- across thousands of kilometres of ecological barriers, like oceans, mountains and deserts.

Birds store fat for use as fuel during their migratory flights. They start to do so well before migration starts. Stored fat is the most economical type of fuel in terms of high oxidizing energy per unit mass. Once the animal is in the right condition to migrate, it may need some further environmental cues to initiate the actual migratory movement. A bird may wait for the right weather conditions or at least the disappearance of the wrong ones, such as fog or very strong winds.

In order to locate its goal at a different position on the earth’s surface, an animal must possess certain sensory systems and have certain decision-making programmes in its behaviour. The ability to take up a particular direction with respect to some feature or property of the environment is called orientation’. The orienting animal is rather like a man who possesses a compass and an instruction to proceed in some particular direction. Animals may orient with respect to objects on the earth’s surface, to the sun, to the stars, to the earth’s magnetic field, to the directions of current flow and so on. Further, they may use more than one of these environmental sources of information at any one time, and may use different ones at different times.

Birds may use continuous flapping flight or soaring flight. Migration of soaring birds attracts much attention among bird-watchers because of spectacular concentrations of these migrants at passages with favourable soaring conditions. The migrants make long detours to avoid having to use flapping flight over the sea, and the sites most famous for soaring bird migration in Europe, Falsterbo, Gibraltar and the Bosphorus, are situated at minimal sea crossings. Migrants’ coasting behaviour probably is part of their ‘strategy’ to exploit winds and conserve energy at the same time. Coastal migration occurs mainly under opposed and cross-winds, while migrants usually fly across the coast and depart over the open sea with following winds. Due to differences in friction, winds generally are stronger over the sea than over land. Migrants minimize the headwind force by following the coast, where they can use local topography and vegetation to gain additional protection from the wind. Furthermore, over the sea they will be exposed to wind drift, and under certain cross-winds it is beneficial to follow a coastline, some distance in the direction of their goal, rather than to take a direct route over the sea.

Some birds travel by numerous short flights, each of about three to ten hours duration. Many species depart either by day or by night, whenever weather becomes favourable. Flying with a small load of fat is advantageous, since the labour costs for carrying the extra fat can be high. However, there are also drawbacks involved in migrating by numerous short flights: the birds have to refuel often and find suitable resting sites on their route.

Some species undertake enormous non-stop flights. A regular migration route from North to South America directly over the Western Atlantic Ocean, a distance between 3,000-4,000 km, is used by many species. Just to mention some examples of long, nonstop flights over oceans other than the Atlantic: there are some geese that fly almost 4,000 km across the Pacific Ocean from the Alaska Peninsula to the South Californian coast; two species of New Zealand cuckoos fly about 3,500 km to the Solomon and Samoa islands; passerines, bee-eaters and Amur falcons travel almost 3,000 km of the Indian Ocean between India and East Africa.

Many migrants fly between Europe and Africa across the Mediterranean Sea and the Sahara desert in one single non-stop flight, lasting at least 40 hours in the autumn, when winds are generally favourable, and 60 hours in spring when they are less favourable.

Honey buzzards migrate by cross-country soaring over a distance of 7000 km from Europe to tropical Africa without refuelling. Common buzzards cover a distance of 10000 km between East Europe/West Siberia and South Africa by soaring migration. Many young Manx shear waders, in their autumn flight from Britain to Brazil, cover almost 0,000 km without refuelling. Various water species wintering in South Africa cover the total distance to their high Arctic Eurasian breeding grounds, about 13,000 km, in four or so long flights.

Arctic terns are transglobal travellers on a scale unequalled by any other migrant. Some annually commute between the Northern and Southern pack ice, crossing the equator twice. The round trip must be at least 40,000 km (25,000 miles).

Low temperatures seem not to prevent birds from migrating at high altitudes, The highest migrants over Puerto Rico experience temperatures of about –12 C, and over Switzerland migration is perfectly regular at altitudes with temperatures around -IC to -15 C. In fact, the birds’ capacity to fly under conditions of low temperature and low oxygen pressure is so great as to be hardly credible: On 9 December 1967, a radar controller in Northern Ireland reported an echo at high altitude moving south over the Hebrides. The radar height finder indicated an altitude between 8,000 and 8,500 m. The pilot of an aircraft in the vicinity was asked by radio to make a course deviation to pass near the position of the echo. In doing so, the pilot reported a flock of about 30 swans at just over 8,200 m. The observation probably refers to whopper swans, which are known to migrate, sometimes in the middle of the winter; from Iceland to Britain. At 8,200 m, the temperature was -48 C and very strong northerly tailwinds blew at this altitude.

Although, according to some radar studies, probably 90% of birds travel below 2,000 m, two small flocks close behind each other of curlew-like birds were observed at I 0,000 m above sea level.

These observations, indeed, are extraordinary. One wonders, are such enormous flight altitudes merely exceptional or do some birds regularly migrate that high?

We tend to think of events which happen regularly, as ‘normal’ or ‘ordinary’. But are they really? Very often, we do not realize the dimensions of the events. Hundreds of millions of birds migrate from one place to another. Tens of thousands of birds start to fly together. They fly very long distances together and land and feed together. Many of them have never flown that long, nor been to those places. But still they are able to fight against all the odds and finish their journey How do they know when to start? How do they recognize the landscape? How do they decide which route to take? How do they find their way to their destination? How do they know where they can find food? How come so many birds can act together? How and when did they learn to use the sun and other stars, or the earth’s magnetic field, for navigation? How did they acquire the necessary means in the first place?

There are numerous such questions to ask and very often, it is not easy to find the right answers. When the other migrating animals are taken into account, the questions get more complicated and more challenging Hundreds of millions of animals migrate twice a year covering thousands of kilometres. We must ask questions which start with how and why and not take ‘ordinary’ events for granted.