# 7.7: Properties

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- 25462

Chaos is not precisely defined in mathematics, but it occurs where:

- Population dynamics appear to oscillate erratically, without outside influence.
- The population has an unlimited set of different patterns of oscillation, all for the same parameter values.
- The slightest change in the number of individuals can change the population from one pattern of oscillations to any other pattern.

It is not important that you learn all the details of chaos. The important scientific point here is that complexity can arise from simplicity. Complex behavior of something in nature does not imply complex causes of that behavior. As you have seen, as few as two lines of computer code modeling such systems can generate extremely complex dynamics. The important biological point is that populations can oscillate chaotically on their own, with no outside influences disturbing them, and that their precise future course can be unpredictable.

Chaos and randomness in deterministic systems were discovered by mathematician Henri Poincaré late in the 19^{th} century, but the knowledge scarcely escaped the domain of mathematics. In the 1960s, meteorologist Edward Lorenz discovered their effects in models of weather, and in the 1970s theoretical ecologist Robert May made further discoveries, publishing a high-profile paper that landed in scientific fields like a bombshell. The details of chaos were then worked out by a large assortment of mathematicians and scientists during the last quarter of the twentieth century. The discrete-time logistic equation examined in this chapter is now designated by Ian Steward as number sixteen of seventeen equations that changed the world.

**Fi****gure** \(\PageIndex{1}\). The final six of seventeen equations that changed the world, as designated by Ian Steward.