Chaotic heteroclinic networks as models of switching behavior in biological systems

The paper presents a clear qualitative mechanism: a local vertical perturbation F = F0 + g supported near a fundamental domain S on a heteroclinic connection Γ1 creates a nontrivial intersection WuF(p1) ∩ WsF(p2), induces up/down branching according to whether points are pushed above/below Γ1, and yields sensitive dependence; dwell times near p2 shorten for larger |height| and lengthen for smaller unstable eigenvalue λu. These points are explained graphically and narratively but not proved rigorously; transversality and formal chaos (e.g., horseshoes) are not established, and key steps are asserted as “easy to check” . The candidate solution gives a substantially more mathematical sketch using the Stable Manifold Theorem and Hartman–Grobman and even proposes Franks’ lemma to tune λu; however, its IVT step incorrectly measures height to Γ1 while concluding intersection with WsF(p2), which requires distance to WsF(p2) or an argument via the paper’s explicit description WsF(p2) = [x2,p2] ∪ ⋃n≥0 F−n([x2,x3]) to locate x* with F(x*) ∈ [x2,x3] ⊂ WsF(p2) . Both accounts convincingly convey the idea and the monotone dependence of dwell times, but each leaves gaps at the level of a complete proof.