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 [-ε_{0},ε_{0}]
for 0 < h ≤ h_{0} and
-α_{0} ≤ α ≤
α_{0}. 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(h^{p}) 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(h^{p})-closeness: only a weakly singular
estimate O(h^{p}⋅ ln
1⁄α) is proved. Nevertheless, numerical
experiments suggest that this estimate is sharp for our
particular construction of J. Uniform
O(h^{p})-closeness in the α >
0 case is proved under an additional assumption on the normal forms.