An introduction to bifurcation theory

Bifurcation theory attempts to explain various phenomena that have been discovered and described in the natural sciences over the centuries. When a parameter on which the behaviour of the system depends changes slightly, one would expect that this should not make a big difference. However, if the value of the parameter crosses some critical value, sudden 'qualitative' or topological changes of the
behaviour of the system may occur. In this case we say that bifurcation occurs.

I am going to give an introduction to bifurcation theory as applied to ODE and show several examples of different types of bifurcations in the simplest cases. Then I will proceed to the so-called Andronov-Hopf bifurcation - this is one of the first types of 'bifurcation' that comes to mind when one hears the word. The work done by these two scientists undoubtedly deserves attention and also this particular type of bifurcation is encountered in a number of physical problems.