Discrete VHCs for Propeller Motion of a Devil-Stick using purely Impulsive Inputs

The paper proves N-periodicity of the discrete zero dynamics for φ ∈ {±π/2} and Δθ* = 2π/N under Assumption 1 by working directly with the implicit update (58) and its specialized form (94), showing ω2 = ω1 (95) and then pairing steps to force equalities like ω_{N/2+3} = ω_{N/2} (via (96)–(101)) and, for odd N, ultimately ω_{N+1} = ω1 (117)–(122). Crucially, Assumption 1 rules out the nontrivial factor ω_a ω_b = P sin(mΔθ*) that would otherwise obstruct these equalities. Hence (θ_{k+N}, ω_{k+N}) = (θ_k, ω_k) holds for the two prescribed phases. The candidate solution hinges on an incorrect “explicit” update ω_{k+1} = (P sin θ_k)/ω_k (mis-solving the quadratic), which leads to a false anti-periodicity claim for N ≡ 3 (mod 4). The correct explicit quadratic solution is given in (85), and the paper’s implicit-method proof does not rely on branch choices. Therefore, the model’s conclusion conflicts with the paper’s correct argument. See (58)–(60) for the DZD, the periodicity proof steps (90)–(101) and (117)–(122), and Theorem 1.