Discrete Hopf bifurcation for Runge-Kutta methods

Keywords: Bifurcation problems Hopf bifurcation Computational methods for bifurcation problems Attractors and their bifurcations a b s t r a c t An investigation into how well real bifurcations in the family of dynamical systems are approximated as the step-size varies is carried out. The preservation of bifurcation structures and stability under numerical simulations is discussed. In addition, the behaviour of numerical solutions generated by a Runge–Kutta method applied to a dynamical system whose analytical solution undergoes a Hopf bifurcation is investigated. Hopf bifurcation results for the numerical solution are presented and analysed. An investigation into how well real bifurcations in the family of dynamical systems are approximated as the step-size varies is carried out. In particular, we prove that Runge–Kutta numerical methods preserve all the real fixed point bifurca-tions of the parameterised family of dynamical systems. It is well known, that one way to investigate dynamical systems is to look at parameterised families. However, what we will do in this work is to study a dynamical system and then treat the step-size as the bifurcation parameter. The treatment of the step-size Dt as a bifurcation parameter is now well established in numerical analysis of dynamical systems. However, the case of numerical approximation of families of dynamical systems, when there are multiple bifurcation parameters, Dt and also any parameters in the ODE, is not well studied. In addition, we consider the stability of steady states when fixed point bifurcations are approximated numerically. We show, that close to a bifurcation point there is no restriction on the step-size. Hence, arbitrarily large step-sizes will produce the correct stability behaviour. Finally, the behaviour of numerical solutions generated by a Runge–Kutta method applied to the d-dimensional system of ordinary differential equations defined by