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Leaves on most trees are arranged in a fractal pattern. This pattern should influence their arrangement on the ground after they have fallen. The wind conditions during their fall (and the flutter nonlinearity) will also play a role in their final arrangement. Nevertheless I don't see how the Lorenz attractor plays any role.

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No, there is no Lorenz attractor there. The trajectory of a particular falling leaf does not influence the trajectory of the next leaf that falls. (They are of course correlated, because they are both heavily influenced by the prevailing winds. But correlation does not imply causation.) Without this feedback, there is no chaos, only randomness.

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The cheap and easy answer to this is that the double pendulum is considered chaotic because it is very sensitive to small perturbations in initial conditions (amongst other things). Showing this mathematically may be difficult (see the Lagrangian formulation for the dynamics), but if one looks at the animations on the Wikipedia page showing the trajectory ...

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Periodic orbits are possible for low energies and very high energies (eg both limbs performing circular motion around the fixed point). In between there is a transition stage (quasi-periodic motion) towards chaos, followed by a transition back to periodic motion. I have not come across any statement of initial conditions defining the boundaries between ...

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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 ...

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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 ...

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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 ...

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Or am I completely off track? Yes, you are. Some hints to get you back on track: Is there something else in your solution that depends on the initial conditions? What type of dynamics does this system exhibit? What Lyapunov exponent do you expect from this? There are some limits in the definition of the Lyapunov exponent.

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This is an interesting question. Formally, yes, when you tear the paper, there is sensitive to initial conditions, and that explains why it is very unlikely that two pieces of paper are teared exactly in the same shape on every tear. However, the fact that there is chaos does not imply that something can not be described macroscopically. There is nothing ...

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The way I see it you just need to do the same thing in two dimensions that you did in one dimension for the. Now if $$x_{n+1} = f_1(x_n, y_n)$$ $$y_{n+1} = f_2(x_n, y_n)$$ you need to find functions for $$ln |{\partial f_1 \over \partial x}|$$ $$ln |{\partial f_1 \over \partial y}|$$ $$ln |{\partial f_2 \over \partial x}|$$ $$ln |{\partial f_2 \over ... 0 is the exit position (or vicinity of the exit) considered an attractor? Sort of. Let’s first consider a single pedestrian who wants to exit the room and whose position in the room is (x,y). Let’s further assume that the exit is located at (0,0). Then we may describe the pedestrian’s position with the following differential equations:$$ \dot{x} = ...

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An attractor is defined in phase space. Phase space is the space of all degrees of freedom of your system. So in your example it cannot be a spatial location such as a room exit. Instead you have to imagine how many parameters describe the motion of one person (a lot), then how many persons there are, multiply the two and you will get the size of phase ...

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I think, after reading what´s written above, we must make a distinction between systems that are big and little, like the atmosphere and a single billiard ball. I´s clear that a billiard ball on a table makes a journey that deviates more and more from the path it would have taken hadn´t you give a slightly different direction in velocity. Consider now a ...

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