Discretizing the transcritical and pitchfork bifurcations -- conjugacy results

We present two case studies in one-dimensional dynamics concerning the discretization of transcritical (TC) and pitchfork (PF) bifurcations. In the vicinity of a TC or PF bifurcation point and under some natural assumptions on the one-step discretization method of order $p\ge 1$, we show that the time-$h$ exact and the step-size-$h$ discretized dynamics are topologically equivalent by constructing a two-parameter family of conjugacies in each case. As a main result, we prove that the constructed conjugacy maps are $\mathcal{O}(h^p)$-close to the identity and these estimates are optimal.