Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: one dimensional cases

The paper proves that the tropical discretization x_{n+1}=F_τ(x_n,c)=x_n(x_n+τ f)/(x_n+τ g) preserves the codimension-1 steady-state bifurcations (saddle-node, transcritical, supercritical pitchfork) of the ODE x' = F(x,c)=f−g via derivative-scaling identities at the steady state and shows that a flip bifurcation in τ occurs at τ=κ(x̄,c)=−2x̄/(x̄D+2f) when D(x̄,c)<0 and κ>0, with the first three flip conditions automatic and the nondegeneracy ∂³(F_τ∘F_τ)/∂x³≠0 required . The candidate solution reproduces precisely these results. It derives the same derivative-scaling using H_τ:=F_τ−x=M_τF with M_τ>0, yielding the identical tests and preservation for all three bifurcations, and it re-derives κ and verifies the three “automatic” flip conditions using the same chain-rule structure (with dF_τ/dx=1+Z_τD and ∂_τZ_τ>0 at x̄). Aside from a minor notational ambiguity (using F_{τ,τ} to denote ∂_τF_τ rather than ∂²_τF_τ), the logics coincide. Hence both are correct with substantially the same proof idea.