Difference between stochastic process and chaotic system Can anyone please point out some difference and similarity between stochastic system  and chaotic system?
 A: I think this is an excellent question that maybe doesn't have
a clear-cut answer. Here is my personal attempt.
Technically speaking, I suppose a 'stochastic process' can be
almost any collection of random variables (usually indexed by time).
But the stochastic processes of most interest are those in which some
sort of 'glue' holds these random variables together so that the
long run behavior (as time passes) can be described in some meaningful manner.
IID. As suggested by @Michael a sequence of iid random variables
would be considered a stochastic process. Even though no one
observation in the sequence helps predict others, there are
some interesting rules of behavior described the the Law of Large
Numbers and the Central Limit Theorem (under reasonable conditions).
Markov. Markov processes have a sort of one-step dependence, that may or
may not wear off as time passes. Consider process with only
two states 0 and 1. (a) If the process simply alternates between
0 and 1 at each step in time, starting at state 1 at step 1,
then it is always in state 1 at every odd-numbered step. 
Technically Markovian, but not very interesting. (b) If the
process moves from its current state to the other with probability
1/3 at each step, then the influence of its starting state
soon wears away. All we can say is that in the limit it will spend
half of the steps in 0 and half in 1. (c) A process that starts
in state 0 and moves to state 1 when the first 6 appears on
a repeatedly rolled die, then it will take an average of six
steps to get to state 1 and then stay there forever. 
"Chaotic." I think the terminology 'chaotic' is reserved for processes
in which it is essentially impossible to describe the long
run behavior in any meaningful way knowing the initial state
(and even knowing some of the rules of movement among states).
Traditionally and most simply, coin tossing is often considered 'chaotic' in the sense
that there is no predicting whether the coin will show H or T.
One might try to take into account the speed and spin with
which it is tossed, but very small differences in speed and spin
can determine H or T in a way that is not generally considered predictable. 
We do suppose that a fair coin will land H or T and that it will
show each result half of the time. But relative to the simplicity
of the probability model, coin tossing has a 'chaotic' feel because it does
not seem possible to predict beyond the obvious.
Closer to the main sense of a 'chaotic' process may be weather
and its unpredictability. Specific forecasts about rain are
usually futile more than three or four days in advance. There
are too many tiny factors that can affect whether a storm will
develop, stay coherent, or fade away to make a reasonable
forecast. Even a forecast of 'scattered showers' for tomorrow in my area
tells me essentially nothing about whether I get rain in my garden tomorrow.
Definitions? I'm sure there are texts on stochastic processes that pretend to
give a narrower definition than mine, and that there are books
on chaos theory that will try to do the same for that field.
However, I do not know of any generally satisfactory definitions that 
make a really clear cut distinction. I'm posting this hoping that
someone comes closer to 'mathematics' than to 'intuition' in
making the distinction.
