As we know, one of the important characters of chaos is , being bounded. also we know that it is sensitive to initial conditions. so ,if we write a code to create a bounded random system in matlab with rand() function, would it result as a chaotic system?
No, it would be noise. The point about chaos is that it leads to result similar to noise but for systems which have either no noise, or rather low. In any case, we talk of chaos when sensitive dependence on initial conditions arises through an essentially deterministic mechanism, and not altogether because of randomness,
If you mean the chaos theory (deterministic chaos), then them ARE DIFFERENT by nature. Chaos means that the behaviour of system (no randomness) is not predictable. This is the common feature with random system (a nose), but the rest of body is different.
Deterministic chaos is a mathematical concept while randomness is essentially a philosophical one. (An often met misconception is to consider randomness as a mathematical concept, the wide field of probability theory is merely the mathematical apparatus for a theory of randomness.) I will focus on the mathematical side.
Chaos theory relies on real numbers where the notion of limits is well defined, and chaotic motions are, very simplified, those not tending to a limit. By contrast, so-called random number generators (RNG) in computers work on finite sets where the notion of limit is meaningless. Nevertheless, these sets are so big that they may be considered as approximation not only to the set of integers but also to the set of reals when taking quotients. In this sense, RNG may be considered as approximations to dynamical systems over real numbers. More precisely, to chaotic ones since by design their behaviour depends highly on initial conditions. To my knowledge, the RNG used by Mathlab belongs to this kind of "chaotic systems". That the RNG are approximations at best does not matter: inside the computer any simulation of a chaotic system is an approximation.
Two important differences between chaotic systems and RNG remain to be mentioned. First, theory and range of application of chaotic systems is much wider than that of RNG. Second, methods developed to study chaotic systems and indicators to characterize them have rarely, if ever, applied to RNG: it is not known if a specific RNG is a chaotic system or not (what I mentioned above is mainly heuristics). With this in mind, chaotic systems and RNG are very different things.
So, whether the Mathlab code you propose will work like a chaotic system cannot be answered generally. Moreover, if the code ignores the input of the RNG then the output will probably be not chaotic. In any case, methods of chaos theory have to be applied to the resulting system. And finally, whether what you will obtain is "random" (or "noise") is again the philosophical question.
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