On spurious asymptotic numerical solutions of explicit Runge-Kutta methods

The bifurcation diagram associated with the logistic equation υn+1 = aυn(1 − υn) is by now well known, as is its equivalence to solving the ordinary differential equation (ODE) u′ = αu(1 − u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. We investigate, both analytically and computationally, Runge-Kutta schemes applied to the equation u′=f(u), for f(u) = αu{1 − u) and f(u) = au(1 − u)(b − u), contrasting their behaviour with the explicit Euler scheme. We determine and provide a local analysis of bifurcations to spurious fixed points and periodic orbits. In particular we show that these may appear below the linearised stability limit of the scheme, and may consequently lead to erroneous computational results.