Discretizing the transcritical and pitchfork bifurcations -- conjugacy\n results

2014/11/23 by Lajos Lóczi, Lóczi, Lajos
Computer Science · Mathematics · Physics and Astronomy · #37G10 #65L05 #65P30 #65Q30 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1411.6252

openalex publication_date 2014/11/23 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We present two case studies in one-dimensional dynamics concerning the\ndiscretization of transcritical (TC) and pitchfork (PF) bifurcations. In the\nvicinity of a TC or PF bifurcation point and under some natural assumptions on\nthe one-step discretization method of order p\≥ 1, we show that the time-h\nexact and the step-size-h discretized dynamics are topologically equivalent\nby constructing a two-parameter family of conjugacies in each case. As a main\nresult, we prove that the constructed conjugacy maps are calO(h^p)-close\nto the identity and these estimates are optimal.\n

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