This article is from the Nonlinear Science FAQ, by James D. Meiss firstname.lastname@example.org with numerous contributions by others.
(Thanks to Zhen Mei for Contributions to this answer)
A bifurcation is a qualitative change in dynamics upon a small variation in
the parameters of a system.
Many dynamical systems depend on parameters, e.g. Reynolds number, catalyst
density, temperature, etc. Normally a gradually variation of a parameter in
the system corresponds to the gradual variation of the solutions of the
problem. However, there exists a large number of problems for which the number
of solutions changes abruptly and the structure of solution manifolds varies
dramatically when a parameter passes through some critical values. For
example, the abrupt buckling of a stab when the stress is increased beyond a
critical value, the onset of convection and turbulence when the flow
parameters are changed, the formation of patterns in certain PDE's, etc. This
kind of phenomena is called bifurcation, i.e. a qualitative change in the
behavior of solutions of a dynamics system, a partial differential equation or
a delay differential equation.
Bifurcation theory is a method for studying how solutions of a nonlinear
problem and their stability change as the parameters varies. The onset of
chaos is often studied by bifurcation theory. For example, in certain
parameterized families of one dimensional maps, chaos occurs by infinitely
many period doubling bifurcations
There are a number of well constructed computer tools for studying
bifurcations. In particular see [5.2] for AUTO and DStool.