Introduction.  Welcome to the online course "What is Chaos?" We hope you find it a useful and entertaining way to learn about one of the most exciting topics in physical science. In physics, chaos is a word with a specialized meaning, one that differs from the everyday use of the word. To a physicist, the phrase "chaotic motion" really has nothing with whether or not the motion of a physical system is frenzied or wild in appearance. In fact, a chaotic system can actually evolve in a way which appears smooth and ordered. Rather, chaos refers to the issue of whether or not it is possible to make accurate long-term predictions about the behavior of the system.  For four centuries in physics, the laws of physics have reflected the complete connection between cause and effect in nature. Thus until recently, it was assumed that it was always possible to make accurate long-term predictions of any physical system so long as one knows the starting conditions well enough. The discovery of chaotic systems in nature about 100 years ago has all but destroyed that notion. After going through the five lessons of this course, you will understand how and why this is true. In Lesson One, you will learn about the Philosophy of Determinism. This is the underlying belief in physical science that says every cause has a unique effect, and vice versa. In Lesson Two, you will learn about Initial Conditions, which is how physicists refer to the starting measurements of any system. Using the assumed link between cause and effect, the initial conditions are used to make predictions.   In Lesson Three, you'll learn about Uncertainty of Measurement, which is the principle that says that no measurement can be made with infinite accuracy.  In Lesson Four, you'll learn how determinism, initial conditions, and the uncertainty of measurements give rise to Dynamical Instabilities, which to most physicists is a term synonymous with Chaos.  Finally, in Lesson Five, you'll learn about how chaotic motion may give rise to large-scale ordered structures. We hope enjoy the course and find in useful. When you have completed it, you can consult the Suggestions for Further Reading.

Determinism is the philosophical belief that every event or action is the inevitable result of preceding events and actions. Thus, in principle at least, every event or action can be completely predicted in advance, or in retrospect. As a philosophical belief about the material world, determinism can be traced as least as far back as the time of Ancient Greece, several thousand years ago. Determinism became incorporated into modern science around the year 1500 A.D. with the establishment of the idea that cause-and-effect rules completely govern all motion and structure on the material level. According to the deterministic model of science, the universe unfolds in time like the workings of a perfect machine, without a shred of randomness or deviation from the predetermined laws. The person most closely associated with the establishment of determinism at the core of modern science is Isaac Newton, who lived in England about 300 years ago.   Newton discovered a concise set of principles, expressible in only a few sentences, which he showed could predict the motion in an astonishingly wide variety of systems to a very high degree of accuracy. Newton demonstrated that his three laws of motion, combined through the process of logic, could accurately predict the orbits in time of the planets around the sun, the shapes of the paths of projectiles on earth, and the schedule of the ocean tides throughout the month and year, among other things. Newton's las are completely deterministic because they imply that anything that happens at any future time is completely determined by what happens now, and moreover that everything now was completely determined by what happened at any time in the past. Newton's three laws were so successful that for several centuries after his discovery, the science of physics consisted largely of demonstrating how his laws could account for the observed motion of nearly any imaginable physical process. Although Newton's laws were superseded around the year 1900 by a larger set of physical laws, determinism remains today as the core philosophy and goal of physical science.

Initial Conditions. One of the important innovations that created modern science around the year 1500 A.D. was the idea that the laws of the material universe could be understood meaningfully only by expressing physical properties as quantified measurements, that is, in numerical terms and not just in words. The use of numerical quantities to describe the physical world is the reason why the laws of physics must ultimately be expressed as mathematical equations, and not simply as ordinary sentences. For example, although Newton's laws are expressible in words, in order to apply the laws to study a particular system, it is necessary to employ the laws in their form as mathematical equations.  Newton's laws are perhaps the most important examples of dynamical laws, which means that they connect the numerical values of measurements at a given time to their values at a later or earlier time. The measurements that appear in Newton's laws depend on the particular system being studied, but they typically include the position, speed, and direction of motion of all the objects in the system, as well as the strength and direction of any forces on these objects, at any given time in the history of the system. In expressing the measurements appropriate for a given system---whether it be the Solar System, a falling object on earth, or ocean currents---the values of the measurements at a given starting time are called the initial conditions for that system. As dynamical laws, Newton's laws are deterministic because they imply that for any given system, the same initial conditions will always produce identically the same outcome. The Newtonian model of the universe is often depicted as a billiard game, in which the outcome unfolds mathematically from the initial conditions in predetermined fashion, like a movie that can be run forwards or backwards in time. The billiard game is a useful analogy, because on the microscopic level, the motion of molecules can be compared to the collisions of the balls on the billiard table, with the same dynamical laws invoked in both cases.

Uncertainty of Measurements.  One of the fundamental principles of experimental science is that no real measurement is infinitely precise, but instead must necessarily include a degree of uncertainty in the value. This uncertainty which is present in any real measurement arises from the fact that any imaginable measuring device--even if designed and used perfectly---can record its measurement only with a finite precision. One way to understand this fact is to realize that in order to record a measurement with infinite precision, the instrument would require an output capable of displaying an infinite number of digits. By using more accurate measuring devices, uncertainty in measurements can often be made as small as needed for a particular purpose, but it can never be eliminated completely, even as a theoretical idea. In dynamics, the presence of uncertainty in any real measurement means that in studying any system, the initial conditions cannot be specified to infinite accuracy. In the study of motion using Newton's laws, the uncertainty present in the initial conditions of a system yields a corresponding uncertainty, however small, in the range of the prediction for any later or earlier time. Throughout most of the modern history of physics, it has been assumed that it is possible to shrink the uncertainty in the final dynamical prediction by measuring the initial conditions to greater and greater accuracy. Thus, in studying the motion of a rocket, for example, one could know the final position of the rocket ten times as accurately by specifying the initial conditions at launch ten times as accurately. It is important to remember that the uncertainty in the dynamical outcome does not arise from any randomness in the equations of motion--since they are completely deterministic--but rather from the lack of the infinite accuracy in the initial conditions. The unspoken goal of experimental science has been that as measuring instruments become more and more accurate through technology, the accuracy of the predictions made by applying the dynamical laws will become greater and greater, approaching but never reaching absolute accuracy.

Dynamical Instabilities.  Having understood what is meant by determinism, initial conditions, and uncertainty of measurements, you can now learn about dynamical instability, which to most physicists is the same in meaning as chaos. Dynamical instability refers to a special kind of behavior in time found in certain physical systems and discovered around the year 1900, by the physicist Henri Poincaré. Poincaré was a physicist interested in the mathematical equations which describe the motion of planets around the sun. The equations of motion for planets are an application of Newton's laws, and therefore completely deterministic. That these mathematical orbit equations are deterministic means, of course, that by knowing the initial conditions---in this case, the positions and velocities of the planets at a given starting time---you find out the positions and speeds of the planets at any time in the future or past. Of course, it is impossible to actually measure the initial positions and speeds of the planets to infinite precision, even using perfect measuring instruments, since it is impossible to record any measurement to infinite precision. Thus there always exists an imprecision, however small, in all astronomical predictions made by the equation forms of Newton's laws. Up until the time of Poincaré, the lack of infinite precision in astronomical predictions was considered a minor problem, however, because of a tacit assumption made by almost all physicists at that time. The assumption was that if you could shrink the uncertainty in the initial conditions --- perhaps by using finer measuring instruments---then any imprecision in the prediction would shrink in the same way. In other words, by putting more precise information into Newton's laws, you got more precise output for any later or earlier time. Thus it was assumed that it was theoretically possible to obtain nearly-perfect predictions for the behavior of any physical system. But Poincaré noticed that certain astronomical systems did not seem to obey the rule that shrinking the initial conditions always shrank the final prediction in a corresponding way.  By examining the mathematical equations, he found that although certain simple astronomical systems did indeed obey the "shrink-shrink" rule for initial conditions and final predictions, other systems did not. The astronomical systems which did not obey the rule typically consisted of three or more astronomical bodies with interaction between all three. For these types of systems, Poincaré showed that a very tiny imprecision in the initial conditions would grow in time at an enormous rate. Thus two nearly-indistinguishable sets of initial conditions for the same system would result in two final predictions which differed vastly from each other.  Poincaré mathematically proved that this "blowing up" of tiny uncertainties in the initial conditions into enormous uncertainties in the final predictions remained even if the initial uncertainties were shrunk to smallest imaginable size. That is, for these systems, even if you could specify the initial measurements to a hundred times or a million times the precision, etc., the uncertainty for later or earlier times would not shrink, but remain huge. The gist of Poincaré's mathematical analysis was a proof that for these "complex systems," the only way to obtain predictions with any degree of accuracy at all would entail specifying the initial conditions to absolutely infinite precision. For these astronomical systems, any imprecision at all, no matter how small, would result after a short period of time in an uncertainty in the deterministic prediction which was hardly any smaller than if the prediction had been made by random chance. The extreme "sensitivity to initial conditions" mathematically present in the systems studied by Poincaré  has come to be called dynamical instability, or simply chaos. Because long-term mathematical predictions made for chaotic systems are no more accurate that random chance, the equations of motion can yield only short-term predictions with any degree of accuracy. Although Poincaré's work was considered important by some other foresighted physicists of the time, many decades would pass before the implications of his discoveries were realized by the science community as a whole.. One reason was that much of the community of physicists was involved in making new discoveries in the new branch of physics called quantum mechanics, which is physics extended to the atomic realm.

Manifestations of Chaos.  From the first four lessons, you have learned that in a chaotic system, using the laws of physics to make precise long-term predictions is impossible, even in theory. Making long-term predictions to any degree of precision at all would require giving the initial conditions to infinite precision. At the time of its discovery, the phenomenon of chaotic motion was considered a mathematical oddity. In the decades since then, physicists have come to discover that chaotic behavior is much more widespread, and may even be the norm in the universe. One of the most important discoveries was made in 1963, by the meteorologist Edward Lorenz, who wrote a basic mathematical software program to study a simplified model of the weather.  Specifically Lorenz studied a primitive model of how an air current would rise and fall while being heated by the sun. Lorenz's computer code contained the mathematical equations which governed the flow the air currents. Since computer code is truly deterministic, Lorentz expected that by inputing the same initial values, he would get exactly the same result when he ran the program. Lorenz was surprised to find, however, that when he input what he believed were the same initial values, he got a drastically different result each time. By examining more closely, he realized that he was not actually inputing the same initial values each time, but ones which were slightly different from each other. He did not notice the initial values for each run were different because the difference was incredibly small, so small as to be considered microscopic and insignificant by usual standards. The mathematics inside Lorenz's model of atmospheric currents was widely studied in the 1970's. Gradually it came to be known that even the smallest imaginable discrepancy between two sets of initial conditions would always result in a huge discrepancy at later or earlier times, the hallmark of a chaotic system, of course. Scientists now believe that like Lorenz's simple computer model of air currents, the weather as a whole is a chaotic system. This means that in order to make long-term weather forecasts with any degree of accuracy at all, it would be necessary to take an infinite number of measurements. Even if it were possible to fill the entire atmosphere of the earth with an enormous array of measuring instruments---in this case thermometers, wind gauges, and barometers --- uncertainty in the initial conditions would arise from the minute variations in measured values between each set of instruments in the array. Because the atmosphere is chaotic, these uncertainties, no matter how small, would eventually overwhelm any calculations and defeat the accuracy of the forecast. This principle is sometimes called the "Butterfly Effect." In terms of weather forecasts, the "Butterfly Effect" refers to the idea that whether or not a butterfly flaps its wings in a certain part of the world can make the difference in whether or not a storm arises one year later on the other side of the world. Because of the "Butterfly Effect," it is now accepted that weather forecasts can be accurate only in the short-term, and that long-term forecasts, even made with the most sophisticated computer methods imaginable, will always be no better than guesses. Thus the presence of chaotic systems in nature seems to place a limit on our ability to apply deterministic physical laws to predict motions with any degree of certainty. The discovery of chaos seems to imply that randomness lurks at the core of any deterministic model of the universe. Because of this fact, some scientists have begun to question whether or not it is meaningful at all to say that the universe is deterministic in its behavior. This is an open question which may be partially answered as science learns more about how chaotic systems operate. One of the most interesting issues in the study of chaotic systems is whether or not the presence of chaos may actually produce ordered structures and patterns on a larger scale. Some scientists have speculated that the presence of chaos---that is, randomness operating through the deterministic laws of physics on a microscopic level---may actually be necessary for larger scale physical patterns to arise. Recently, some scientists have come to believe that the presence of chaos in physics is what gives the universe its "arrow of time," the irreversible flow from the past to the future. As the study of chaos in physics enters its second century, the issue of whether the universe is truly deterministic is still an open question, and it will undoubtedly remain so, even as we come to understand more and more about the dynamics of chaotic systems.