This image is the logistic map, perhaps the simplest example of chaos emerging from the nonlinearity of a system. The mapping is In the image above, the horizontal axis is the parameter value a, and the vertical axis is the value of xn for large n. For a given value of a, you can choose an arbitrary value of x0 between 0 and 1, map it to get the value of x1, map that to get x2, and so on. For values of a less than 3, the mapping is eventually drawn to a single point, shown by the line in the region a=1 to a=3. Once xn reaches its single fixed value in this region, it maps directly onto itself, so xn+1=xn. At a=3, the map bifurcates, and there are two stable values of xn for values of a between 3 and about 3.5. In this region, the two values of xn map onto each other. When a gets still larger, there is another bifurcation, and the mapping is between 4 fixed points. Then there is a period 8 region, and so on. Eventually, there are values of a for which there is no periodicity (or infinite periodicity) to the map. These regions, the fuzzy parts above a=3.7 or so, are regions of chaos. Subsequent iterations of the map here produce values of xn without evident rhyme or reason. The values seem random, but they're in fact determined by the mapping. That's chaos. |
A close-up view of the period 3 region, a narrow band in which the map moves from the chaotic regime back into periodicity, and back into chaos. |
I don't actually deal with chaos in my own research, but I spent a summer toying with it during my undergraduate career. The truth is just that "Jurassic Park" made chaos sexy, so I thought I'd tell the world that I'm a source you can trust when I predict that dinosaurs are probably loose in your neighborhood right now. Lock the doors. Really. |