One-dimensional Piecewise Smooth Rational Degree Maps.

The paper proves a trichotomy for the piecewise map f, constructing the first-return map F and the induced two-parameter map G(z,λ), then deriving: (i) uniform expansion and hence chaos when ν > p/(p+q) (Lemma 8), (ii) a period-doubling (flip) at z*=p/(p+q) with λ_PD = q (p+q)^{(p−q)/q} ν^{p/q} / (p^{p/q} (1−ν)) when 1 − q (p+q)^{(p−q)/q} / p^{p/q} < ν < p/(p+q), with nondegeneracy verified via K1≠0 and K2<0, and (iii) stability below that threshold; μ_PD and the interval I_M are then expressed explicitly (Theorem 1; Lemmas 9–10; Proposition 12) . The model’s solution mirrors this construction, recovers the same thresholds and bifurcation location, and correctly identifies z* and λ_PD. Minor gaps remain: it checks weaker nondegeneracy conditions and sketches, rather than fully derives, the exact closed form for μ_PD; nonetheless the main claims and mechanisms align with the paper.