Nonlinear systems are those mathematical systems and natural phenomena that are not linear. The study of nonlinear dynamical systems is sometimes called nonlinear science. Strange attractors and limit cycles can only appear in nonlinear systems. Non-Linearity is sometimes used as a buzzword.
Non-Linear, Non-Equilibrium and Non-Elephant
Stanislaw Ulam apparently once remarked: "Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.” (the quote can be found in James Gleick "Chaos: Making a new science", Viking Penguin, 1987 and on page 374 in Campbell et al., "Experimental mathematics: the role of computation in nonlinear science", Commun. Assoc. Comput. Mach. 28 (1985) 374–84)
The study of non-linear systems in physics is in fact like the study of non-elephant systems in biology or zoology. Nearly everything interesting is non-linear. The vast majority of interesting mathematical equations and natural phenomena are nonlinear, with linearity being the exceptional, but important, case. Nonlinear systems are ubiquitous, linear systems are the exception. Linear systems are simple, nonlinear systems are complex. There are many forms of complexity, but only a few forms of simplicity: all linear systems are similar, but nonlinear systems can be quite different from each other.
"Nonlinear system" is a word similar to "Non-Equilibrium System", an ambiguous word for a vague concept. Next to "Non-Equilibrium Systems" comes "Dissipative Structures". The worst phrase of all is "systems far from equilibrium". It does not says how from equilibrium the system is really, or in which direction.