Nonlinear and Chaotic Dynamics


In our day-to-day lives, we become accustomed to devices and objects behaving in a very linear fashion. For instance, if we reach for a volume knob on a stereo or a television, we anticipate that the volume will increase proportional to how far we turn the knob in one direction or the volume will decrease if we turn the knob in the other direction. Such devices are designed to meet that expectation and we come to take linear responses for granted.

Nature, however, has a heightened complexity due to the relatively large number of factors that can affect even what is thought of as relatively simple phenomena. The pitfall of this complexity is most obvious in trying to predict the weather. One might expect that since we are able to measure temperature and pressure, humidity and wind speed, that knowing the weather on any particular day would allow us to predict how the weather will change based on these deterministic factors.

Relatively small uncertainties in measurements, however, will lead to an error in prediction that may grow in time in a nonlinear or exponential fashion. Hence, we can speak with some certainty, based upon the current weather pattern, what the weather is likely to be in an hour or a day, but this is a relatively small time scale and the weather a week or a month from today cannot be accurately predicted.

Imagine the stereo described above but designed a little differently. You turn the volume knob up a little, and the stereo is a little louder. You do so again, and the same thing happens. Now you turn the knob up just a little more and the volume increases by a factor of four, or drops to half of what it just was. Then you turn the volume back down that little bit, and the volume isn't what it was just a moment ago when the knob was in the same position. Such issues are the basis for research in nonlinear and chaotic systems.



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