Numerical solution of Volterra equations undergoing bifurcations

We consider the numerical solution of some Volterra integro-differential equations of the form: \[ y'(t)= g(t)= \int_0^t k(t,s,y(s)) ds \] By careful choice of the original equation we can give an analysis that shows four distinct types of behaviour in the exact solution. The challenge for the numerical methods is then to show that we obtain the correct behaviour in the numerical solution for each of these types of true behaviour. We focus on simple numerical methods and give diagrams that illustrate how well each method performs.