## 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 < *h* ≤ *h*_{0} and - *α*_{0}
≤ *α* ≤ *α*_{0} with
*h*_{0} and *α*_{0} sufficiently
small. Then the quantity *|x - J(h,x,α)|* is proved to be
O(*h*^{p}) small, uniformly in *x* and
*α*, in a small *x*∈
[-*ε*_{0},*ε*_{0}]
neighbourhood of the origin, where *p* denotes the order of the
one-step discretization method.