Well, the 1st idea that comes to mind is coarse graining: Say you have the Lorenz-system.

https://en.wikipedia.org/wiki/Lorenz_attractor

So the system is basicly circling around in one of two circles with about constant frequency. Which of the two circles is choosen after each round only depends on the initial conditions (and in reality on the accuracy of your chosen intergrator), but is so hard for a human eye to predict that it appears to be random. The most time spent during one round in either of the circles would then correspond to a "random" binary sequence. 'Zero' for a circle of your choice, 'unity' for the other.

There was once a scientist in our seminar talking about this exact idea, but I can not remember who it was, since it was already a few years ago.

Now I remember: the one who gave the talk in our seminar was Andrej Shilnikov (also here on Researchgate). I hope his paper might be helpful:

Article Symbolic Quest into Homoclinic Chaos

The "symbolic description" mentioned in the abstract is, if I remember correctly, exactly these "random" words generated from the chaotic dynamics.

Since I have not read the paper, I simply assume that a method for generating these random sequences from the dynamics of the Lorenz attractor or something similar should be described somewhere in this paper. That would be the answer to your question, if I understood it correctly.

The idea of coarse graining is correct, and the fancy (and rigorous) way to do this is to use symbolic dynamics. With a properly chosen partition (such as a generating partition), one can find an equivalent representation of a chaotic dynamical system as a shift operator on an infinite sequence of discrete symbols, which is a symbolic dynamics. With a symbolic dynamics, there is a one-to-one mapping between system states and symbol sequences. With only finite information about an initial condition (i.e., limited precision), the shift will eventually reveal new information about the initial condition that was previously unknown and is practically random. The symbolic representation is a convenient method to identify the new (random) information. In addition, symbolic dynamics allows you to analyze the statistics of the random sequences, ideally by constructing an equivalent Markov model.

An entry point to this could be [Bollt, Lai, Grebogi, "Coding, Channel Capacity, and Noise Resistance in Communicating with Chaos," Phys Rev Lett 79, 3787].

Thank you Drs Patrick and Ned for the resources. I will look into them

The idea is not new but there is no chance to compete with existing and widely used random number generators (RNG). First of all, because of finite computer capacity, any deterministic RNG is either periodic or lets the program crash since memory is exhausted (the latter happens e.g. when producing the digits of $\pi$ by the spigot algorithm, see [J. Gibbons: Unbounded Spigot Algorithms for the Digits of Pi, [email protected]]).

In case of periodically working RNG the first question is its period length,

then questions for uniformity and independence follow.

Among the periodic RNG the probably most simple (and therefore best analysed) one is the Linear-Feedback-Shift-Register approach (see English Wikipedia: LFSR). Here a long logical word of bits (may be much larger than the physical word length) is shifted to the left, the one bit shifted out is XORed with the bits at specific positions and fed back as the new right most bit. If there are $n$ bits then, depending on the choice of bits contributing to the new bit, the period of the bit sequence is up to $p=2^n-1$ (using a common PC, the time before the first period is over may be larger then the time since the big-bang, *n>100* does the job). The auto-correlation function is, obviously, periodic too and takes the value 1 at time lags $k*p$ (because of periodicity) and the value $-1/(2^n-1)$ in between (or the like, I do not remember the precise figure). This means that the bits in the generated sequence are practically independent from each other. A similar effect is obtained when forming 32-bit words out of the sequence . These and many more properties can be proved by embedding the LFSR into an algebraic framework: polynomials over a finite field. Such algebraic support is not available for chaotic systems like the Lorenz' one. Moreover, the auto-correlation function of the Lorenz system and may others (I do not know a counter example) is far from being zero.

Taking all together, chaotic systems used as RNG are statistically much worse than the LFSR (even the LFSR is not the last word), and the algorithm to produce the numbers is dramatically slower. This I meant when I claimed that there is no chance to compete.

Thank you for the insights, Prof. Jansen. However, I was propped to ask the question after reading a host of works that demonstrated successful conversion of chaotic sequences into random number generators for application in cryptosystems. Experts such as Sundrapandian Vaidyanathan, Cristo Volos, Thahm among others have produced several papers on chaotic random number generation using chaotic maps and flows. My interest is on how to get these discrete RNG from flows, specifically. I believe it is useful in engineering applications of chaos.

I cannot answer this question by experience but I can explain what I would do. I would intersect the attractor's range by a Poincare plane (not necessarily a plane, a curved but still smooth surface may be more appropriate, this depends on the attractor in question). Observing the trajectory's crossings through the plane (always in the same direction, ignore the backward crossings) gives a map. The times between crossings are well separated, so the chaos can "develop", the effect is strengthened when only every tenth or so crossing is observed. It may be wise to split the plane into numbered parts, the number of the part actually hit gives the random number. Here, the Lorenz system is really nice because of the attractor's obvious separation into a "left wing" and a "right wing" yielding a bit sequence. I have no idea how to obtain a meaningful partition in case of other, less symmetric attractors (maybe, this is the kernel of your question).

So far, I discussed the chaotic systems simulated by a computer. If the system is hardware instead of software then replacing the crossings of the Poincare plane by stroboscopic observations may do the job (but in hardware I have even less experience than in software).

Anyway, strong correlations between the crossing points or part numbers remain, I would never apply this approach.

Thank you Prof, for your comments.

Well, the numbers generated by any chaotic system in digital domain will not be random anymore due to quantization. Though if some properties are satisfied they can be called pseudorandom and can be used for applications. The period length and statistical properties of chaotic sequences matter a lot. Using the output of any chaotic map as a sequence of random numbers is pretty easy. You just have to define your scale limits in which you want your random numbers be generated. For instance, random number between 0 and 1 or random numbers between 0 and 256. After generating sequences from chaotic maps all you have to do is scale them. If you have multi-dimensional map you can also concatenate them before scaling. Again please take into account the quantization error. This following work can help you more.

https://link.springer.com/article/10.1007%2Fs11071-019-05112-

https://rdcu.be/bKhar