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 ${\cal{O}}(h^p)$-close to the identity and these estimates are optimal.Comment: 38 pages, 9 figure

A model function for polynomial rates in discrete dynamical systems

Hüls T. A model function for polynomial rates in discrete dynamical systems. APPLIED MATHEMATICS LETTERS. 2004;17(1):1-5.ln this paper, we construct a one-dimensional map with a nonhyperbolic fixed point at zero for which the orbits converging to zero and the solution of the associated variational equation can be determined explicitly. We extend the construction to parameterized systems where the fixed point undergoes bifurcations. Applications are indicated to heteroclinic orbits that connect a hyperbolic to a nonhyperbolic fixed point with one-dimensional center manifold. (C) 2004 Elsevier Ltd. All rights reserved

A model function for polynomial rates in discrete dynamical systems

In this paper we construct a one-dimensional map with a non hyperbolic fixed point at zero for which the orbits converging to zero and the solution of the associated variational equation can be determined explicitly. We extend the construction to parameterized systems where the fixed point undergoes bifurcations. Applications are indicated to heteroclinic orbits that connect a hyperbolic to a non hyperbolic fixed point with one-dimensional center manifold