Is a falling leaf an example of a chaotic system? Let´s assume is a wind still day in autumn. When a little change is made in the initial motion of a leaf at the time it falls off a tree, the resulting path of motion of the leaf is very different from the path that would develop if these changes wouldn´t have been made. 
All the leafs though reach the ground within a maximum radius (wich is a function of the height of the tree) caused by chance effects. Can we, because we know that the leaves land within a certain area, still say that a falling leaf is a chaotic system? Or do we have to consider an infinite high tree, and consider the combined system of the air and the falling leave?
 A: Yes. With the qualification that chaos describes the behaviour of an ideal (continuous?) mathematical model, whereas leaves and the air through which they fall are real, I think it is a chaotic system.  Leaves falling from the same place but with a small change in orientation can land in very different places, and intermediate orientations do not necessarily land in between. As the difference in orientation is changed by smaller and smaller amounts (ad infinitum), the path of connected landing points becomes more convoluted. 
I think that increasing the height of the tree will also increase the radius of landing sites on the ground, without limit. 
A: It depends on your exact definition of chaos:


*

*We certainly have a strong sensitivity to initial conditions (butterfly effect), which is the one property of chaos everybody seems to agree upon.

*We do not have topological mixing.

*The falling to the ground is only a short-lived transient compared to the non-chaotic lying on the ground. So, at most, we have a chaotic transient.
To get something that is chaotic by all definitions (I am aware of), we would have to extend the system a little bit, for example:


*

*look at the entire tree;

*allow the tree to re-grow leaves;

*somehow deal with the leaves that reached the ground and letting their fate influence the system. For example, we could have a literal feedback process by having the rotting leaves act as a fertiliser that makes leaves on the respective side of the tree grow a tiny bit faster.

A: Chaos is typically phrased as a sensitivity to perturbation in initial conditions (amongst other important things things).  You can have a statistical distribution describing the final destination of leaves in general, when the path taken by any individual leaf is deemed chaotic.
As an example, consider common strange attractors.  Its easy to see that there is a statistical distribution in locations where a particle ends up, but predicting where any one particle ends up with any degree of accuracy is impossible after enough distance in orbit around a strange attractor.
