Фокусные особенности и круговые биллиарды с потенциалом Гука

The paper proves that for the circular billiard with repulsive Hooke potential k<0, H=(ẋ²+ẏ²)/2 + (k/2)(x²+y²) and F=xẏ−yẋ form an integrable (piecewise-smooth) system; (0,0) in the (h,f)-plane is a focus–focus singular value, and on the n-sheet book Ω_n the monodromy label equals n, yielding a torus-bundle monodromy matrix [[1,0],[n,1]] and a semilocal focus–focus singularity with n pinches. These appear explicitly as Theorem 1 (monodromy label n) and Theorem 2 (leafwise homeomorphism to a focus–focus singularity with n critical points) in the uploaded text . The paper also details the Hamiltonian S^1-action generated by F and its compatibility with the gluing across sheets . The model solution reaches the same conclusions via a different route: it computes monodromy as the first Chern number of the S^1-bundle induced by the F-flow on the boundary of a small saturated neighborhood and sums local fixed-point indices (+1 per focus–focus fixed point), obtaining n and hence the same monodromy matrix, and it invokes the standard semilocal classification of focus–focus singularities. This is consistent with the paper’s results, and the linearization/eigenvalue check at the equilibrium agrees with the paper’s computation of focus–focus type . The model adds a plausible heteroclinic description of the singular fiber that the paper does not spell out, but it does not contradict the paper. Overall: same claims, different proof techniques.