Biquadratic Nontwist Map: a model for shearless bifurcations

The paper’s appendix defines the same integrable Hamiltonian H(x,y) = −ay + a(1+ε) y^3/3 − aε y^5/5 + (b/2π) cos(2πx) and prescribes the equal-H criterion at the relevant hyperbolic fixed points to locate separatrix reconnection. Using the fixed points z±1=(0,±1), z±2=(0,±1/√ε), z±3=(1/2,±1), z±4=(1/2,±1/√ε) and the convention that x ranges over ℝ, the paper derives b*1 = (4πa/3)(1−ε/5) and b*2 = [2πa(1−5ε+5ε^{3/2}−ε^{5/2})]/(15ε^{3/2}). The candidate solution evaluates H at exactly these points, imposes the same equalities, and recovers the identical formulas. The supplementary hyperbolicity check in the model (via the Hessian) is consistent with the paper’s stability discussion for 0<ε<1. Thus, both are correct, with essentially the same method, and the results match the paper’s equations (24)–(25) and (27)–(28) and the Hamiltonian (26) definition. Citations: H definition and method (), fixed points (), convention on x (), b*1 (; ), b*2 (; ), stability for 0<ε<1 (; ).