Preprint des Projektes: DFG Forschergruppe Spektrale Analysis, asymptotische Verteilungen und stochastische Prozesse

Numerische Approximation und Spektrale Analysis unendlich-dimensionaler Dynamischer Systeme

05-016 Lajos Loczi.
Conjugacy in the discretized transcritical bifurcation

The present work can be considered as another case study - analogous to our earlier preprint [1] - in the direction of discretizing one-dimensional ordinary differential equations near non-hyperbolic equilibria. This time the hyperbolicity condition is violated due to the presence of a transcritical bifurcation point. The main aim is to show that the dynamics induced by the time-h-map of the original continuous system and that of the discretized one are still locally topologically equivalent, meaning that there exists a conjugacy between the corresponding phase portraits in the vicinity of the equilibrium. Besides the construction of a conjugacy map J(h, ⋅, α), the important point is that we also estimate the distance between J(h, ⋅, α) and the one-dimensional identity map.
In the first part of the paper, we derive normal forms for the time-h-map of the ordinary differential equation and its discretization near a transcritical bifurcation point at bifurcation parameter α = 0 in one dimension and with discretization stepsize h > 0. We assume that the discretization method preserves equilibria. We will see that it is sufficient to construct a conjugacy between these normal forms.
In the second part, J(h, ⋅, α) is constructed for 0 < hh0 and - α0αα0 with h0 and α0 sufficiently small. Then the quantity |x - J(h,x,α)| is proved to be O(hp) small, uniformly in x and α, in a small x∈ [-ε0,ε0] neighbourhood of the origin, where p denotes the order of the one-step discretization method.