Dynamics of the Tyson-Hong-Thron-Novak circadian oscillator model

For the two‑equilibria regime of the slow–fast THTN system, the paper proves: (i) in L1M/MR1 there is a stable node on L1/R1, a saddle–node on M, no periodic orbits on x ≥ 0, global convergence to the node except along the saddle–node’s center manifold, and infinitely many heteroclinic connections; (ii) in MM there is an unstable node and a saddle–node on M with a unique heteroclinic; (iii) in L0M/MR0 there is a homoclinic loop near a canard cycle occurring precisely at v = vci(δ) = v0 + Ki δ + O(δ^{3/2}), with Ki = (κi,3 + Ai/4)(D1ψ2)^2(c0 + (xi − φ(xi))^2)/D11ψ1, and the loop exists iff κi,3 + Ai/4 < 0; crossing vci(δ) yields an unstable canard cycle with the stated direction. These are exactly the paper’s statements (Theorems 5.2–5.3, Lemma 5.2 and the proofs around them), including the normal form near canard points and the invariant formulae for vci(δ) and Ki . The candidate solution reaches the same conclusions, but its global no‑cycle/convergence argument invokes planar competitive‑systems theory (Hirsch) rather than the paper’s Fenichel–based invariant manifold construction and attractor argument. Aside from this proof‑strategy difference (and minor regularity details at x = 0), the candidate’s claims, classifications, and vci(δ)/direction results match the paper.