Discretizing the transcritical and pitchfork bifurcations -- conjugacy results

TL;DR

This paper investigates how discretization methods preserve the qualitative structure of transcritical and pitchfork bifurcations in one-dimensional dynamics, establishing conjugacy results that quantify the approximation accuracy.

Contribution

The authors prove that under certain conditions, discretized bifurcations are topologically equivalent to the exact ones, with conjugacy maps close to the identity at a rate of ${ m O}(h^p)$, which is shown to be optimal.

Findings

Discretized dynamics are topologically equivalent to exact dynamics near bifurcation points.

Constructed conjugacies are ${ m O}(h^p)$-close to the identity, with optimal estimates.

Results apply to one-step discretization methods of order $p \\ge 1$.

Abstract

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.