Chaos is not a random process, although it may look like one. If I am given a set of observations, is it possible to determine if the observations are generated by a random process or if they are generated by a chaotic process?
What properties or characteristics of the observations can be used to make that determination if it is possible?
To build on user304539's answer: what you want to know is whether starting in the same initial state $\mathbf{x}_0$ again will move you to the same next state $\mathbf{x}_1=\mathbf{f}(\mathbf{x}_0)$ (the system is deterministic) or to some other state $\mathbf{x}_1+\mathbf{y}$ (the system is indeterministic/noisy - $\mathbf{y}$ doesn't depend on $\mathbf{x}_0$). In practice, since you cannot get a perfect re-run, the question is whether the difference between realizations starting close tends to be small or big. Note that we are ideally talking about a short time between state 0 and 1: at late times chaos and randomness are indistinguishable.
One way of checking this is to make a return map of the system: plot the points $(\mathbf{x}_0,\mathbf{x}_1)$. This is obviously simpler for scalar states rather than vector states since you get a 2D plot, but you can often just select some simple projection function $p(\mathbf{x})$ that projects the state onto a single variable. If the system is a deterministic function of state (and later unpredictability is due to chaos) you will get a plot of $\mathbf{f}(\mathbf{x})$. If it is random, you will just get a point cloud. And if it is somewhere in between, you will get a fuzzy shape.
Here is a return plot of iterates of the logistic map $x_{n+1}=\lambda x_n(1-x_n)$ for $\lambda=3.96$ where I plot $(x_n,x_{n+1})$ (leftmost pane). It is a simple parabolic function, although the points $x_n$ are jumping around chaotically. In the right three panes I add normally distributed noise of S.D. 0.01, 0.1 and 1 to the $x_n$ (clamping them to stay in [0,1]). For the strongest noise there is no real pattern, the system is indeterministic. In the intermediate cases there is a fuzzy shape: the system has a mix of deterministic dynamics (which would be chaotic if noise-free) plus random noise.
One thing worth noting in this example is that even adding a tiny bit of noise makes the attractor shape different - the parabola does not touch 0 or 1 in the noise-free case, but does when subjected to noise. Deducing the "true" deterministic behaviour that would happen without noise can be very hard. In this case you can just fit a simple function and when the fit is used for iteration you will get the right dynamics. But noise can make the dynamics drift away and explore parts of state space that would normally be inaccessible - indeed, without the clamping to [0,1] the iterates would run off to infinity in this case, making the return plot useless.
Random motion of a collection of particles (or a single one) is one for which all particles (or one) show a behavior that is unpredictable (it could be used as a random number generator). Say the motion ao air particles involved in air pressure.
Chaotic motion is predictable but highly sensible to initial values, which makes it hard to predict. Say the motion of particles involved in the weather.
In the weather system, there is a chaotic pattern superimposed on the random pattern. If you consider them separately you can see that the random motion shows no pattern, while the chaotic motion does, and a highly variable one. This (developing) pattern is super sensitive on a change in conditions. If the sun's intensity is reduced a bit, a much different pattern can appear (if the reduction in the sun's power lasts long enough and is big enough). Different from the pattern that would be there id the power would not be reduced.
The problem may come down to the temporal resolution and accuracy of your measurements. In the ideal case one could plot the dynamical variables in a phase space. Although the trajectory of a chaotic system would never repeat on itself, it would be confined within a given volume of that phase space - the 'strange attractor'. Very often, one may characterize this attractor by the dynamics of the system.
Truly random behaviour would follow no trajectory at all but this would be difficult if not impossible to determine unless you could make the temporal resolution of your measurements arbitrarily fine. Of course, there is no paradigm for truly random behaviour in classical physics. Even in quantum physics, the amplitudes determining the probabilities evolve deterministically.
What may appear as random behaviour might be unknown outside influences on the system.
Is not the definition of a chaotic system is that it is indistinguishable from random?
One demonstration of a chaotic system I saw had what was a large number of bouncing balls all in a lossless system with each ball from left to right bouncing at a slightly higher frequency. At the beginning they all bounced in a kind of moving wave. Then the wave started to break up and then it suddenly looked completely random. If one were to isolate any one ball then we could characterize the motion easily. If we did not know which ball we were looking at, or we tried to make some system wide determination of the position of the balls then it would take considerable effort to prove anything deterministic and the system is therefore indistinguishable from random.
Proving a chaotic system is not random can be so difficult that we do not try.
A real world example I can think of is the use of analog computers to predict tide patterns. These computers go back to at least World War II. With careful observations of the tides people can build a model to input into these analog computers. These computers would have some certain number of inputs and the more inputs the more time into the future it could accurately predict the tides. At some point the tides broke from this model and they'd have to make new observations to build a new model.
Presumably if the computer had a knob for every real variable in the tides, and there was a long enough series of observations to accurately adjust the knobs, then the computer could predict the tide indefinitely. In reality though something new would come along to upset the system enough that it became no different than random again.
Perhaps that is the test of a chaotic system vs. random. Can you build a model to predict future events on any time scale?
