On bifurcation from infinity: a compactification approach

The candidate’s analysis reproduces the five-regime attractor description and heteroclinic network in the paper’s Theorem 2.4, including the two pitchforks at λ=σ*1 and λ=σ*2, the escape of ±u1 and ±u2 to infinity as λ↗σ1,σ2, and the induced semiflow at infinity with equilibria ±ϕ∞1, ±ϕ∞2 and heteroclinics ±ϕ∞2→±ϕ∞1 (see Theorem 2.4 and Fig. 2.8 ). The model adds a clean coercivity argument via the Steklov trace matrix and a strict Lyapunov (Rayleigh quotient) on S∞ for the projected flow Ut=(I−U⊗U)LλU, complementing the paper’s hyperplane-projection proof (cf. (2.34) and Lemma 2.3 ). A minor typographical slip in the paper states regime (i) as λ∈(0,σ*1) instead of (−∞,σ*1); the body of the paper and Lemma 2.1 use the correct interval (−∞,σ*1), which matches the model’s correction . Overall, both arguments are consistent and correct; the methods differ in emphasis (Rayleigh/Sturm vs. hyperplane linearization at infinity).