A system is not linear, or nonlinear, if it does not satisfy the superposition principle: its response at a given place and time caused by two or more stimuli is not necessarily the sum of the responses which would have been caused by each stimulus individually.
Similarly, a problem is nonlinear if it cannot be broken into a sum of mutually independent sub-problems: its various components/aspects interact with each other so as to render impossible their separation for solving the problem step by step or in blocks.
With systems/problems that are nonlinear, the analytical approach (dividing into parts to make the problem tractable) is no longer sufficient and it must be complemented with a holistic, global approach to the problem as a whole.
Figure taken from P.Magrassi, Difendersi dalla complessità, Franco Angeli, Milano 2009, pag. 87
Imagine the study of an animal population, aimed at modelling with a mathematical equation the dynamics of the population over time depending on the availability of food.
If there are predators for that animal, then a linear model turns out simplistic and inadequate: the population of preys becomes a function of the predators’ population but, in turn, the latter will expand and contract based on the availability of preys. The “preys – predators – food” system is intrinsically nonlinear: none of its components may be studied in isolation from the others. And indeed, the Lotka-Volterra equations are a classical example of simple nonlinear model of an ecological situation.
Such models, and of much greater complexity, are common today in fields like electronics, optronics, avionics, chemistry, biology, ecology, economics and many other fields.
However, the relevant mathematical equations are seldom solvable, and the problems can be treated only via numerical computer simulations. This is the reason why the study of complex dynamical systems, even if they have been known and marginally studied since the XIX century, has developed since the advent of computers.
The heart of complexity
While there are scientific aspects of complexity (such as computational complexity) that are not necessarily related to non-linearity, the most spectacular and popular manifestations of complexity, namely deterministic chaos and emerging behavior, are indeed due to non-linearity.
Surprisingly, this fact is not captured in most “pop” complexity literature. In Wikipedia, for example, as of 27 July, 2009 non-linearity was absent from all relevant definitions, with the adjective linear appearing only 1,518 words through the “complexity” article, 269 words through “complex systems” and 218 words through “complex system” (two entries with the essentially same name tell a lot about the reigning confusion).
(NB: I personally tried and fix the Italian Wikipedia, it.wiki, in mid-July and I’ll be curious to see what follows).