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Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

Oliver Stein

Abstract. Discretization of autonomous ordinary differential equations by numerical methods might for certain step sizes generate solution sequences not corresponding to the underlying flow - so-called "spurious solutions" or "ghost solutions". In this paper, we explain this phenomenon for the case of explicit Runge-Kutta methods by application of bifurcation theory for discrete dynamical systems. An important tool in our analysis is the domain of absolute stability, resulting from the application of the method to a linear test problem. We show that hyperbolic fixed points of the (nonlinear) differential equation are inherited by the difference scheme induced by the numerical method while the stability type of these inherited genuine fixed points is completely determined by the method's domain of absolute stability. We prove that, for small step sizes, the inherited fixed points exhibit the correct stability type, and we compute the corresponding limit step size. Moreover, we show in which way the bifurcations occurring at the limit step size are connected to the values of the stability function on the boundary of the domain of absolute stability, where we pay special attention to bifurcations leading to spurious solutions. In order to explain a certain kind of spurious fixed points which are not connected to the set of genuine fixed points, we interprete the domain of absolute stability as a Mandelbrot set and generalize this approach to nonlinear problems.

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