Discretizing the transcritical and pitchfork bifurcations -- conjugacy results
For researchers in numerical dynamics, this provides rigorous optimal error bounds for conjugacies in discretizing two common bifurcations.
The paper proves that near transcritical and pitchfork bifurcation points, the exact and discretized dynamics are topologically equivalent via conjugacies that are O(h^p)-close to the identity, with optimal estimates.
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 ${\cal{O}}(h^p)$-close to the identity and these estimates are optimal.