A deterministic system is a system in which a finite set of fixed laws governs the transition of one state of the system to another, a state being one instance of the entire system with all dynamic variables specified (e.g. positions and momenta). The laws determine precisely how the system evolves in time or throughout space. In other words, given the laws and the initial state (or any specified state) of the system, all other states of the system could be correctly determined. The problem with such a system would still be in measuring precisely the variables.
“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 a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”
~Pierre-Simon Laplace
The alternative to a deterministic system is an indeterministic system. An indeterministic system is a system in which there is no finite number of laws which precisely specifies the time evolution of a system. Another way of saying this is that the states of a system are disconnected, in that there is no true dependence of any state on another given state.
A deterministic system can also be called a dynamical system. However, sometimes a dynamical system is considered a more ‘real’ version in which determinism is just a very close approximation. Dynamical systems can have randomness as long as the randomness does not appreciably change the system’s behavior (i.e. the system would appear almost exactly the same if that randomness could be removed somehow).
A dynamical system can be either linear or nonlinear. A linear system is one in which the slight change of a system’s state will only cause changes which are proportional to that change. In a nonlinear system, changing the initial state does not create proportional alterations of the later states. Within a very short duration of time, the system may be unrecognizable from what it would have been if the initial condition wasn’t changed. Take note that when I speak of the initial conditions being changed; a more practical interpretation is that the experimenter imperfectly measured the initial conditions (the change being the difference between the measurement and reality).
Given perfect measurements of a system’s state, the later state of the system can be calculated perfectly for both linear and nonlinear systems. In reality, however, we are rarely afforded with this benefit. Let us say that we are 0.0000000001 meters off in measuring the initial state of a particle in our system. In a linear system this would not make a huge difference. Measuring a later state of the particle would yield a result that is not much more than 0.0000000001 meters off of where the particle would really be.
The problem arises with nonlinear systems. Take the same example as before, where we imperfectly measured our particle’s position by 0.0000000001 meters. Unlike a linear system, it is very likely that in a nonlinear system we could still figure out where the particle will be at some point in the future. The actual future state would be far different from the future state which we calculated.
Another way of making this distinction is that accurate and precise measurements are exponentially more important when dealing with nonlinear systems.
A finite number of variables specifies the state of each particle in the system. A useful way to represent a system is with a phase space. A phase space maps all possible states of a system's motion. A phase space basically represents all possible states a system can take within an n-dimensional grid. A phase space has the same number of dimensions as variables necessary for specifying the system’s state (# particles X number of variables to specify state of each particle). Basically, every degree of freedom is an axis of the entire grid. The portion which is 'colored in' is the phase space. The unshaded parts represent combinations of the variables which the system cannot be in. One point in the n-dimensional phase space is specified by n coordinates (one for each variable).
Here's a picture of a particular system (the Van der Pol oscillator) represented as a 2-D phase space:
(3, -2) is a state of the system which never naturally occurs. (3, 1), however, looks as though it is a state which the system can take. Order of the coordinates does matter. I've taken this ordering to be (x-axis, y-axis). If you still don't understand phase space completely, don't worry. Pictures of attractors in Part 2 will help (plus the video at the bottom of this site and the wikipedia description).
An orbit is a phase-space representation of a sequence of states in time. There are two types of orbits. A periodic orbit is one that repeats the past behavior after a certain fixed period of time (its period). An aperiodic orbit is an orbit where the close repetition of a previous state does not remain close for any appreciable duration of time. It is a nonlinear in that sense. The future cannot be precisely determined if the initial measurement is not 100% correct.
Aperiodic orbits exhibit what is known in chaos theory as sensitive dependence. This can be pictured as saying that small changes in the state of the system will cause very large changes in the overall time evolution of the system. Another way of looking at it is that there are many other orbits which approach the aperiodic orbit, but almost none of them will remain nearby as time progresses. These other orbits can be thought of as the theoretical prediction of motion of the imperfect measured particle (the imperfect measurement being at the point where it approaches the aperiodic orbit). The butterfly effect is the original name given to sensitive dependence by Edward Lorenz.
“Predictability: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?”
~ Edward Lorenz
Chaos theory, then, is the study of nonlinear dynamical (deterministic) systems or the study of systems with aperiodic orbits. The main point of all this is that non-perfect measurements of the state of a chaotic system can’t be used as accurate determiners of the future state of that system. A more practical way to figure out the future of a chaotic system is to just wait until that future moment is the present, and measure it. This ensures the highest amount of accuracy. A full chaotic system is one in which most or all orbits the system can take are aperiodic. A limited chaotic system is one in which most orbits are periodic or nearly periodic, with a few special orbits being aperiodic.
A good documentary about chaos theory:
http://video.google.com/videoplay?docid=-7171768157657288194#
Pendula of all sorts seems to exhibit chaotic behavior:
A cool simulation:
http://www.myphysicslab.com/pendulum2.html
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