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What is Chaos?

In physics the term chaos has a more specific meaning than the same word as it is used in everyday English. Two notions are connected to chaos; one is the notion of a deterministic system and the other is the notion of unpredictability

As physicists we try to explain things by first making certain abstractions about the world as we experience it and then trying to find mathematical models which describe these abstractions. For instance, when we discuss the motion of earth around the sun, we care little about the color of earth and whether it has life on it or not. In predicting earths motion it works actually extremely well to view both earth and sun as points in space, to each of which we assign the corresponding mass. So in this abstraction the only relevant features are the mass of a heavenly body and it's relative position in three dimensional space. Then, with for instance the classic gravitational force law due to Issac Newton, we can describe the motion of these points in three dimensional space. The solution of Newton's equations describes the motion of earth around the sun in time.

Newton's laws are deterministic. If we know precisely the position and mass of all relevant planets, moons, asteroids, etc. at a certain point in time (the initial time), then - according to Newton's and as well Einstein's theory - we can predict the position of earth for all times. The specification of the planets initial positions is called initial condition of the system. Aside from the relative positions which we observe in reality, there are in principle any number of possible initial positions, actually infinitely many. Each of the infinite number of possible initial conditions has exactly one unique future associated with it, the solution of the corresponding equations. This lead to the idea, that if the world is really fundamentally deterministic, e.g. governed by some fundamental deterministic law, then an entity with huge (unlimited ?) computational power could solve these equations and predict the future from the knowledge of the worlds current state. In this view everything, including all my decisions would be predetermined. Aside from all the philosophical arguments which can be raised around this issue, even in physics not all models physicists come up with are deterministic. For instance it is often very useful to use stochastic equations. Deterministic systems are in a certain sense to be seen in contrast to stochastic ones. In stochastic systems we in principle are unable to say what exactly will happen in the future, because we can only talk about the probability of a certain outcome. The main point is that chaos is deterministic! To each initial condition there is one unique future, which we can compute by solving the appropriate equations.

So why then unpredictability? The reason that one can talk about unpredictability in deterministic system is that it is principally impossible to know the initial condition of a system with infinite precision. For nonchaotic systems this caveat does not matter much, because a slight error in the specification of the initial state of a system means that our prediction of the future will be only slightly wrong. For chaotic systems on the other hand the initial uncertainty grows exponentially fast, so that after a very short time it is as large as the possible motion of the system, that is we cannot make any useful prediction.

An example: Consider a satellite in geosynchronous orbit (appr. 36 thousand kilometers above earth), which means it takes the satellite exactly 24 hours to go around earth once with the consequence that it hovers above a specific spot on earth. The problem of one satellite circling earth is approximately a two body problem - we neglect the moon, sun, etc.- which is not chaotic. This implies that if we, for instance, miss the target position and put the satellite one meter further away from earth than intended, then it will still orbit earth and it'll take about 2 milliseconds longer than intended. We see, the consequences are minor, this problem is not sensitive to the initial condition. If the problem were strongly chaotic than missing the intended orbit by one meter could have fatal consequences, e.g. the satellite could go on an awkwardly complicated orbit and not return to earth in years or it might crash into earth. If we were not able to measure the position of the satellite well enough, then this initial uncertainty could, in a matter of days, be amplified so that we would not be able to say whether some days later the satellite would land on your roof or circle nicely around earth. As I said, the motion of geosynchronous satellites around earth is not strongly chaotic. However, if we want to send a satellite to moon, then this is a three body problem (moon, earth and satellite) which for certain parameters does have chaotic solutions. With enough precision on the measurement of the satellites position one can actually use chaos to save fuel. The idea is to shoot the satellite into an orbit around earth and to fire the rockets of the satellite only when it is close to a point in space where a nearby trajectory (read: solution to the appropriate equation) will lead it away from an orbit around earth and along a rather complicated orbit to moon (see J.D.Meiss). This fuel-efficient way to reach the moon relies on the property of chaos that minor changes in the initial position result in very different orbits, the same property which makes long term prediction impossible

Chaos is a property of some solutions of deterministic equations. The defining feature is the practical long term unpredictability of these deterministic systems. The possible existence of these solutions was not appreciated for a long time, mainly because these solutions can usually only be studied with a computer.

For physicists chaos is exciting because very simple mathematical equations were found to have incredibly complicated solutions (chaotic ones) which we think might reflect some of the complicated patterns and rich behaviors we observe in the real world. Knowledge about chaotic mathematical equations gives us therefore a new tool to describe the apparently complicated reality with simple mathematical models. On the other hand, the discovery of chaos shattered the belief that once we found and wrote down these models (or laws) we automatically would have predictive power only limited by our ability to solve the equations. It turns out our predictive power is in many cases not restricted by our computational abilities but is actually more severely limited by our inability to measure the current state of a physical system with high enough precision.