A complex system is typically defined as a system composed of many interacting components whose collective behavior cannot be easily inferred from the behavior of the individual parts. The whole behaves in ways that are not predictable from the sum of its parts.
For those driven by time dependent dynamics.
Why do we put noise in complex systems and how do we do it (white / Poisson / colored / fractional etc..)? what's the motivations?
The main motivation is to better reflect reality, where noise is unavoidable — both because virtually no system is perfectly isolated from the rest of the universe and as a way to account for imperfections of the model.
Depending on what you want to model and/or the limitations of your approach, a number of noise profiles (white, colored, etc.) can be used, often with a tunable amplitude. Mathematically it'll often take the form of an extra term or factor which is random or, in a computer simulation, pseudo-random. This term must be introduced in a way that preserves key aspects of the model. For instance, in a conservative system you might want your noise to not change its total energy.
As we learnt in statistical mechanics, a natural way of describing many-component physical systems is through a statistical description. For example, rather than following the dynamics of all $10^{23}$ molecules of a gas, it is more informative to look at the average properties of such ensemble of molecules, that end up being related with thermodynamic quantities like temperature, pressure and so on.
Physicist's analysis of complex systems builds up in this approach of statistical mechanics. The idea is that microscopic degrees of freedom evolve in an stochastic manner and one seeks extracting average/ensamble properties that provide information at the macroscopic level (of the system as a whole).
The origin of fluctuations in complex systems depend on the particular phenomena under study and on the level of description of the model. Two typical frameworks are diffusion and pure jumping processes. On the one hand, diffusion (SDEs with white/colored noise) emerge when the description of the system is done at a mesoscopic level. The noise in this description represents our ignorance of the particular set of microscopic events, which are coarse-grained into the noise term. On the other hand, jumping process (Poisson noise) is used when one tries to model at a finer-microscopic description. The noise here represents the intrinsic randomness of the evolution of the system at microscopic level. Van Kampen's system expansion is a technique that allows deriving mesoscopic descriptions (diffusions) from microscopic ones (pure jumping processes)
Introduction of typical noise of input data is routinely used in the numerical weather prediction (NWP) in ensemble prediction runs.
The divergence of chosen predicted parameters between runs is tracked and analyzed. This gives the picture about confidence intervals of predicted weather parameters at the given initial synoptic situation and how these intervals increase with time.
See also The ECMWF Ensemble Prediction system (PDF)
ECMWF = European Centre for Medium range Weather Forecast, also the their global NWP model.
