Important Concepts in Chaos Theory

chaos,chaotic,dynamical systems,complex systems,definitions,concepts,terms,dynamics,behavior,period doubling,nonlinear,Butterfly Effect,strange attractors,fractal,fractals,bifurcation,Lyapunov exponents,turbulence,poincare map,universality,self similarity,scale invariance

Definitions of Important Terms in Chaos Theory

Note: Below, we provide intuitive definitions to allow readers with
a non-mathematical background to grasp the main concepts of chaos.
     A system is a set of objects that are governed by a precise set of rules. It is a very general concept. A physical system is one that occurs in nature. Some examples are a collection of billiard balls on a pool table, the weather, a fluid in a heated jar, a flea population housed in a container that is understudy by a ecologist, and the planets of the solar system. In the case of the weather and a fluid, the objects are gas and liquid molecules. A mathematical system is one described by one or more equations. A well known example is

population(t+dt) = A * population(t) * (MaximumPopulation - population(t))
where population(t) is the population at time t and dt is a time interval, and where A and MaximumPopulation are parameters that may be selected to certain values. Mathematical systems are often models in the sense that they are designed to approximate physical systems. The above equation is an insect population model such as an ecologist might use.

     Phase space is the set of all possible states or situations of a system. In the example of billiard balls, all the possible placements of the balls on the pool table constitute's the system's phase space.

     Computer simulation uses computers to reproduce examples of the dynamics of a system. By dynamics, we mean the motion and behavior of the objects as time elapses.

     Parameters are numbers that allow one to vary the rules that govern a system.

     A dynamical system is a system that evolves with time through deterministic equations. In the first of the above examples, the billiard balls and their motion constitute a dynamical system. Newton's laws of motion provide the dynamics.

     A complex system is a dynamical system that exhibits complicated behavior. Its underlying description may possibly be simple (the population equation above is such an example).

     Nonlinear is an adjective that applies to a system to indicate that output responses are not linearly related to input changes. In a linear system, doubling an input doubles the change in the output. An example of a linear relation is the amount of work a person can do versus the time that the person works. If the person works twice as long, he or she can accomplish twice as much. Linear systems are always exactly solvable. A nonlinear system may or may not be solvable. If it is solvable then it cannot be chaotic because it would be predictable.

     In linearizing a system, one approximates a nonlinear system by a linear one.

     Chaos is structured random behavior of a non-linear, complex, dynamical system. The behavior over long-time scales is (1) unpredictable, (2) seemingly random but not arbitrarily so, (3) sensitive to initial conditions, and (4) characterized by a strange attractor that is often a fractal. Chaotic behavior is different from random behavior in that it is not completely random and the strange attractor governs its structure.

     Turbulence is the chaotic flow of a fluid.

     Unpredictability means that one cannot forecast in detail the evolution of the system.

     The definition of random is completely unpredictable, arbitrary and highly varying.

     Sensitivity to initial conditions means that small changes at a particular time lead to large differences later.

     The Butterfly Effect is another term for sensitivity to initial conditions. It originates from the unpredictability of weather: In certain situations, it is believed that a butterfly flapping its wings creates a small disturbance in one particular location that can eventually produce a large change in the weather throughout the world.

     Unstable means not stable. When a small change is implemented in a stable system, the system returns to its original state. An example is a flat bottom jar: If one tilts it slightly then the jar rocks back and forth and soon comes to rest in a vertical position. In an unstable system, a small change leads to a very different behavior. A pencil balancing on its tip is unstable because perturbing it by the slightest amount makes it falls. In many cases, a system is stable in certain directions and unstable in others.

     Lyapunov exponents indicate the rate of divergence of nearby trajectories. Such motions deviate exponentially rapidly and the coefficient in the exponential is the Lyapunov exponent. In other words, in a system with at least one large Lyapunov exponent, there is extreme sensitivity to initially conditions, while in a system with small Lyapunov exponents, nearby situations initially deviate more slowly. There is a Lyapunov exponent for each independent direction.

     An attractor is a region in phase space to which the dynamical motion approaches at long times. In the above example of billiard balls, the six holes or pockets of the pool table can be thought of as an attractor.

     A strange attractor is the attractor for chaotic behavior. It is often a fractal.

     A fractal is a set of points that is of fractional dimension. For example, a curve that is highly irregularly in shape may have a dimension that is effectively greater than one, or a surface that has many holes in it may have a dimension that is less than two. Fractals are self-similarity at small scales.

     A set is self-similar if it looks very much the same when it is magnified.

     The property of a quantity remaining the same when magnified is called scaling. A system that scales is called scale-invariant. A set that is scale-invariant is self-similar.

     Periodic behavior is non-chaotic since the system returns to its original state after a certain time, at which point it repeats what it previously did. The period is the minimum time it takes for the motion to repeat.

     Non-periodic motion is motion that does not repeat.

     A bifurcation is a sudden change in a system that occurs when a parameter is slightly modified. Sometimes it refers to the specific situation in which a system is asymptotically periodic and the periodic behavior undergoes doubling. For example, a system may oscillate about two states. Then a parameter is increased and the system oscillates about four states, two of which are close to one of the original states and two of which are close to the other original state.

     The period-doubling road to chaos refers to a case in which a series of bifurcations occurs at an ever increasing rate as a parameter is varied to a limiting value. At the limiting value, chaos transpires.

     Universality in chaos is the idea that very different systems can exhibit the same type of chaos. An example is the class of systems that follow the period-doubling road to chaos.

     Renormalization group theory is the theory that governs a system that is scale-invariant. It is used to establish universality classes.

     The Poincaré map is the projection onto a plane of the motion of a body in three-dimensional space. It simplifies the analysis of a trajectory by reducing the dimension of the problem by one.

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