# Complexity

Complex dynamical systems are "systems of systems" whose state evolves over time. Some dynamical systems are chaotic (simple deterministic rules produce unpredictable behavior). Others are complex (many interacting parts) and may have stochastic aspects.

## References

Abraham, R. *Complex Dynamical Systems*. http://www.ralph-abraham.org/articles/MS%2398.GST2/neu2.fm.pdf

http://www.scholarpedia.org/article/Bifurcation

http://amath.colorado.edu/faculty/jdm/faq.html

http://en.wikipedia.org/wiki/Dynamical_systems

http://www.scholarpedia.org/article/Dynamical_Systems

http://www.cscs.umich.edu/~crshalizi/notebooks/chaos.html

http://en.wikipedia.org/wiki/Non-linear_system

## J^T Thoughts

First, let's get the terminology straight. There's lots of confusion. Here's definitions: [NOTE: This working draft needs corrections to properly cite sources for the following quotes. (References are to the right.) The following is not my wording!]

### Bifurcation

A bifurcation of a dynamical system is a qualitative change in its dynamics produced by varying parameters.

### Chaos

Effectively unpredictable long time behavior arising in a deterministic dynamical system because of sensitivity to initial conditions.

### Complex system

Complex systems are spatially and/or temporally extended nonlinear systems characterized by collective properties associated with the system as a whole--and that are different from the characteristic behaviors of the constituent parts.

### Deterministic process

A process that has only one allowed next state given a current state and input; no randomness is involved.

### Dynamical system

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.

### Fractal

A geometric figure or natural object is said to be fractal if it combines the following characteristics: (a) its parts have the same form or structure as the whole, except that they are at a different scale and may be slightly deformed; (b) its form is extremely irregular, or extremely interrupted or fragmented, and remains so, whatever the scale of examination; (c) it contains "distinct elements" whose scales are very varied and cover a large range." (Les Objets Fractales 1989, p.154)

### Nonlinear

In mathematics, a nonlinear system is one whose behavior can't be expressed as a sum of the behaviors of its parts (or of their multiples.) In technical terms, the behavior of nonlinear systems is not subject to the principle of superposition. Linear systems are subject to superposition.

When a system is linear, people examining it can make certain mathematical assumptions and approximations about its behavior, allowing for simple computation of results. For instance, the height of a column of water poured into a glass is a simple function of the volume of water poured in, along with the diameter of the glass, making it easy to calculate the height of various possible volumes of water.

In nonlinear systems these assumptions cannot be made. Since nonlinear systems are not equal to the sum of their parts, they are often difficult (or impossible) to model, and their behavior with respect to a given variable (for example, time) is extremely difficult to predict. When modeling non-linear systems, therefore, it is common to approximate them as linear, where possible. The weather is famously non-linear, where simple changes in one part of the system produce complex effects throughout.

### Stochastic process

A process that, given a current state and input, has multiple possible next states with probabilities.