Discrete Hopf bifurcation for Runge-Kutta methods (Q2379059)

The article is devoted to the question how well real bifurcations in the family of dynamical systems are approximated as the step-size is varying. It is proved that Runge-Kutta numerical methods preserve all the real fixed point bifurcations of the parameterized family of dynamic systems. The treatment of the step-size \(\Delta t\) as the bifurcation parameter in the case of the numerical approximation of families of dynamical systems in the presence other bifurcation parameters, \(\Delta t\) and any ones in the ordinary differential equation (ODE), now is not sufficiently studied. Therefore the author investigates the behaviour of numerical solutions generated by a Runge-Kutta method applied to the \(d\)-dimensional system of ODEs \(\dot{y}=f(y,\rho)\), \(y\in \mathbb{R}^d\), \(\rho \in \mathbb{R},\) with continuous mapping \(f:\mathbb{R}^d \times \mathbb{R}\to \mathbb{R}^d, \) whose analytical solution undergoes a Hopf bifurcation. Hopf bifurcation results for the numerical solution are presented and discussed.