Visualizing Attractors of the Three-Dimensional Generalized Hénon Map

Section IV of the paper proves that for the 3D Hénon map L(x,y,z)=(δz+α−σy+x^2, x, y) with δ>0, every bounded orbit lies in the cube |x|,|y|,|z|≤κ with κ=1/2(|σ|+δ+1+sqrt((|σ|+δ+1)^2+4|α|)), via a three-case monotone-escape argument that uses the forward/backward recurrences (15)–(16) and identifies κ as in (14) . The candidate solution obtains the same bound by a concise supremum argument on the scalar recurrence xt+1=xt^2−σxt−1+δxt−2+α, yielding 0≥M^2−(|σ|+δ+1)M−|α| and hence M≤κ; this matches the paper’s κ and correctly concludes that all bounded orbits are in the κ-cube. The model’s proof is valid and simpler, while the paper also establishes the stronger monotone blow-up outside the cube .