Preprint of the project: DFG research group "Spectral analysis, asymptotic distributions and stochastic dynamics"Numerical approximation and spectral analysis of infinite-dimensional dynamical systems05-016 Lajos Loczi.
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.
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