317Chapter 28Non-Linear Systems, Chaos,
Complexity and Emergence
Most systems in nature are inherently nonlinear and can only be
described by nonlinear equations, which are difficult to solve in a closed
form. Non-linear systems give rise to interesting phenomena such
as chaos, complexity, emergence and self-organization. One of the
characteristics of non-linear systems is that a small change in the initial
conditions can give rise to complex and significant changes throughout
the system. This property of a non-linear system such as the weather is
known as the butterfly effect where it is purported that a butterfly
flapping its wings in Japan can give rise to a tornado in Kansas. This
unpredictable behaviour of nonlinear dynamical systems, i.e. its extreme
sensitivity to initial conditions, seems to be random and is therefore
referred to as chaos. This chaotic and seemingly random behaviour
occurs for non-linear deterministic system in which effects can be linked
to causes but cannot be predicted ahead of time.
Most of the simple systems that physicists have considered up to the
time of the latter half of the twentieth century were simple linear systems
giving one the impression that linear systems were the norm and non-
linear systems the exception. In fact the opposite is true. Most systems in
nature are actually non-linear and chaotic. A system as simple as three
bodies problem interacting with each other through gravity is non-linear
as was discovered by Poincaré towards the end of the 19th century. He
was the first scientist to discover a chaotic deterministic system.
Before the availability of the computing power of the last 50 years the
mathematical description of non-linear systems was subject to very
limited numerical procedures. With increased computing power,
however, scientists have been able to identify new structures and forms
of organization within non-linear systems such as fractal structures,