## Complexity

While chaos is the study of how simple systems can generate complicated behavior, complexity is the study of how complicated systems can generate simple behavior.
Complex systems are spatially and/or temporally extended nonlinear systems characterized by emergent collective properties.

### Cellular Automata

(John von Neumann, S. Ulam, and S. Wolfram)

#### What they are

• grid of 0-1 cell values (N cells total)
• simple rule (about altering cell status) applied to each cell, simultaneously - iterated in time
• rule depends on status of neighbors

#### Game of Life

NxNxN...xN matrix of cell values (that can be real or imaginary values) that change over time according to 1) the status of 'neighbors' (or 'connected' cells, or some external condition) and 2) some rule, which can be global (as here) or local. This 'system' 'evolves' over time.
##### Rules
• NNeighbors>3 ->0 (overcrowding)
• NN<2 -> 0 ("loneliness")
• NN=2 retain current state
• NN=3->1 (staying alive or birth)
##### These system have "attractors"
• static states - "fixed point"
• periodic states - "limit cycles"
• nonperiodic states ("chaotic" or "strange attractor")

There are variations in rules, kinds of entities, ...
Vivarium and LifeLab are examples (free or shareware).
##### Genetic Algorithms
[John Holland - Artificial Intelligence (AI)]
• Approach: GA cell values: not species member, but genotypes.
• Goal: system that learns.

### Self-organized complexity (Per Bak)

Sandpile Model
• NxM cells, each with height h ("grains")
• one cell chosen randomly and h increased by 1
• iterated
• if cell h>z, 1 "grain" passed to each neighbor (updated)
• 'avalanche' have size s (number of cells updated)
• continue the iteration and count frequency, D(s) of each s
• Result: log[D(s)] is power law of log(s)
When this occurs it illustrates self-organized criticality.

For a review, perhaps you can look at the Glossary