||Chaotic systems have two definitive properties: (1) they are deterministic, and (2) their long-term behavior sensitively depends on their initial conditions. Edward Lorenz, the father of modern chaos theory, summarized chaos as being present when "the past determines the future, but the approximate past doesn't determine the approximate future." The most familiar depiction of chaotic behavior is the so-called "butterfly effect," in which a very small perturbation of the global weather system (the flapping of a butterfly's wings in Argentina) results in a very large change to that same system (the genesis of a hurricane in Texas three weeks later). Of course, it is clear that not all butterfly flaps spawn hurricanes (thankfully!), so the central problem of chaos theory is the precise mathematical modeling of this sensitive dependence relationship, and the determination of when very small changes are likely to have very large effects. Many important natural systems exhibit chaotic dynamics under certain circumstances; in addition to the global weather system, the global climate, social systems (like the economy), and biological systems (like the human brain) can sometimes exhibit chaotic dynamics. The understanding and modeling of chaos is an important part of understanding complex natural systems.