Piecewise closed form expression of Rössler-like trajectories

The paper defines a piecewise trajectory using eqs. (1)–(5): a logistic carrier for radii, a per-revolution radius R(t) built from a sigmoid plus Gaussian bumps (eq. 3), X,Y via rotation at uniform angular speed (eq. 2), and a height Z(t) given by an exponential envelope tied to r_i (eqs. 4–5) . The paper asserts it focused on expressions with zero derivative at the extremes to avoid discontinuities in the piecewise construction, but with the explicit choices in eq. (3), the end derivatives are only exponentially small—not exactly zero—so C1 joins are not guaranteed across seams; similarly, the Z expression is not periodic and generally produces an O(1) discontinuity at revolution boundaries because Z_i(1)≪Z_{i+1}(0) (with θ reset) . The non-self-intersection claim is presented as a graphical inference invoking a “non-intersection theorem,” rather than a proof; the construction as written needs hypotheses and a direct argument to ensure injectivity in 3D . The candidate model supplies the missing rigor: it proves per-revolution injectivity (θ strictly monotone on [0,1)), excludes cross-revolution intersections by using a radius-dependent height factor, and prescribes standard C∞ “flat-end” windows to make X,Y,Z globally C1. It also correctly explains the inheritance of period-p behavior from the logistic carrier, consistent with the paper’s intent and with classical logistic-map dynamics (May 1976). Net: the paper’s argument is incomplete (missing regularity conditions and a proof of non-intersection), while the model’s solution is correct and fills the gaps (modulo a minor fix: the example Φ(θ)=sin^4(θ/2−π/12) does not vanish at θ=0,2π; use Φ(θ)=sin^4(θ/2) or another 2π-periodic flat-end envelope).