Discretizing the transcritical and pitchfork bifurcations – conjugacy results

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 , we show that the time- exact and the step-size- 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 -close to the identity and these estimates are optimal.