Preprint des Projektes: DFG Forschergruppe Spektrale Analysis, asymptotische Verteilungen und stochastische Prozesse

Numerische Approximation und Spektrale Analysis unendlich-dimensionaler Dynamischer Systeme

04-022 Lajos Loczi.
Construction of a conjugacy and closeness estimates in the discretized fold bifurcation

Our present work is a case study of numerical structural stability of flows under discretization in the vicinity of a non-hyperbolic equilibrium. One-dimensional ordinary differential equations with the origin undergoing fold bifurcation at bifurcation parameter value α = 0 are considered together with their discretizations. In a neighbourhood of this equilibrium a conjugacy is constructed between the time-h-map of the solution flow of the ODE and its stepsize-h discretization of order p ≥ 1 in the limiting case h → 0+.
As we have shown in [4], the conjugacy problem between the original ODE and its discretization can be reduced to the construction of a conjugacy between the corresponding normal forms, which in turn amounts to solving a one-dimensional functional equation depending on the two parameters h and α. A solution – being the required conjugacy map – is now obtained by applying the technique of fundamental domains.
The main emphasis in this work is put on estimating the distance between the conjugacy map J(h, ⋅, α) and the identity on [-ε0,ε0] for 0 < hh0 and -α0αα0. Since the origin is a fold bifurcation point, we can assume that both normal forms possess two fixed points for α < 0 which merge at α = 0 then disappear for α ≥ 0. For α ≤ 0, we show that |x - J(h,x,α)| is O(hp) small, uniformly in x and α ≤ 0, further, that this closeness result is optimal. For α > 0 however, we are currently unable to establish uniform O(hp)-closeness: only a weakly singular estimate O(hp⋅ ln 1⁄α) is proved. Nevertheless, numerical experiments suggest that this estimate is sharp for our particular construction of J. Uniform O(hp)-closeness in the α > 0 case is proved under an additional assumption on the normal forms.