The world around us abounds with chaotic systems: dripping taps (when a certain dripping rate is reached the dripping becomes irregular, which can be seen in this old but very entertaining video, around 30 minutes; the straw that breaks the camel's back [around 6 minutes] is, by the way, not an example of chaos but of criticality), waterfalls, double pendulums, bouncing billiard balls on a pool table, the onset of turbulence and of course the weather system.
Now assume these systems don't contain potential energy sources to be released (like dams with a huge amount of water behind them, a fully blown ballon, the already mentioned camel, or humans; it's obvious that in examples containing cases like this very different histories develop if, for example, an explosive device that is coupled to these dams which will explode (thereby releasing the potential energy of the water) if some conditions vary very little from the non-explosive state then clearly the histories are very different; or if a person sees the doors of the train he had to catch in time close before his eyes then his day will completely change; I think these are both examples of criticality, like the straw that...; so here's no chaos involved).
If we consider a double pendulum, we can't say (to determine if it's subsequent movement is chaotic) to vary a tiny part of the double pendulum, like changing the movements of a number of atoms somewhere in the pendulum (in proportion with changing a small part of the global weather system).
I can't see that there are potential energy sources in the weather system to be released by a tiny change of a puff of air. So isn't it fair to say that in determining if the weather system is chaotic we have to change the whole weather system by a small amount (give all the air molecules the same tiny change in phase space)?