Dynamic Consistency of Discretization
|Original title:||Dynamic Consistency of Discretization|
|Type:||Article in Proceedings|
Numerical methods for solving differential equations transform continuous dynamical systems into discrete ones. We naturally ask whether the dynamics is preserved and for parameter-dependent systems, whether the discretization preserves also bifurcations of equilibria, limit cycles or basins of attraction. We can study these problems using bifurcation theory and find the critical value of the time-step as a parameter of the discrete system. We find for example that all Runge-Kutta method of the odd order undergo the flip bifurcation of the stable (continuous) equilibria. We usually assume that the dynamics is preserved under discretization of sufficiently small step of the numerical method, but this assumption seems to be excessive.