The mathematics of chaos has been popularized through the notion of the butterfly effect: the possibility that a large storm in New England may be caused by a butterfly wing flap in China. There are problems with this simple notion. These problems are keys to recognizing the difference between the models of chaos and the application of these ideas in most real complex systems.
To explain the difficulty, consider the many butterflies that might be responsible for a storm. Which butterfly was responsible? The answer is any or none of them depending on the specifics of the conditions.
It is reasonable to say that most of the time, nothing a butterfly will do can be connected in any way to the existence of a storm in New England. The reason is that particular conditions are necessary in order for sensitivity to exist. For example, hurricanes can be found in New England, but they occur during hurricane season.
Moreover, when special conditions exist that make the weather sensitive to small disturbances, then they are sensitive to many possible small disturbances. This means that a single butterfly is not responsible. Simply stated, if one butterfly changes its flapping, this might cause a change in whether or when a storm occurs. This effect could, however, be reversed by another butterfly changing its flapping. In this case, which one was responsible? There is no direct and unique link between a particular butterfly and the storm. Thus, we have to distinguish between the notion of sensitivity and causation. The storm behavior may be sensitive to the butterfly, but is not caused by it.
Finally, it is not established experimentally that sensitivity in weather extends as far as a butterfly flap of a wing. Even though mathematical models suggest that such things might in principle happen, there is no confirmation of this effect. Nevertheless, the lack of predictability of the weather and the existence of amplification due to feedback is clear in weather patterns. Understanding where and when and the degree of sensitivity is an active area of research.
These ideas can be better understood using a multiscale description of complex systems. There are many more behaviors at fine scales than at large scales. This means there are many more parameters describing the butterflies and relatively few parameters describing the storms. This is different from the usual low dimensional chaos models where a single parameter (real number) is used to describe the entire behavior of the system.