Some examples of non-linear systems and characteristics of their solutions

Abstract

Complexity science is often seen as the science of emerging non-linear phenomena. In this paper the authors discuss some emerging aspects of non-linear solutions in physics. These solutions owe their elegance and simplicity to the complex non-linear structure of the equations, a structure which is dictated by the symmetries of physics. A central theme in these non-linear solutions is that the magnitude of the driving term (or the initial cause in more mundane language), is of little influence on the final solution. In linear approaches one would normally exploit the smallness of the source term by constructing solutions order by order. The non-linear solutions have a very different nature and cannot be constructed by such perturbative means. In contrast to certain other applications in complexity theory, these non-linear solutions are characterized by great stability. To go beyond the dominant non-perturbative solution one has to consider the source term as well. The parameter freedom in these equations can often be reduced by self-consistency requirements. The attempt is to assess a possible role of this type of solutions in general complexity theory. Being stressed is the possibility that the complexity of the equations is beneficial rather than detrimental towards the solution of these non-linear equations, as long as this complexity reflects fundamental aspects or principles in the description of the system.