The canonical dm³ toy system lives on the contact 3-manifold with contact form \(\alpha = dz - r^2 d\theta\): \(\dot r = r(1-r^2) + 2(r-1)e^{{-z}}\), \(\dot\theta = 1\), \(\dot z = r^2 - 2(r-1)^2 e^{{-z}}\). (One source chapter transcribes the coupling term as \(e^{{-r}}\) rather than \(e^{{-z}}\) — the executable certification script certify_rstar.py uses \(e^{{-z}}\), and that’s treated as authoritative here since it’s the version that actually runs and reproduces \(r^*=0.77594059\).) The limit cycle is \(\Gamma = \{{r=1\}}\), period \(T^*=2\pi\).
Theorem μ.1 (Stability radius from Lyapunov exponent, proved). Linearise the \(r\)-equation at \(r=1\): set \(r=1+\varepsilon\), expand to first order: \(\dot r|_{{r=1+\varepsilon}} \approx -2\varepsilon + O(\varepsilon^2)\), so \(\dot\varepsilon = -2\varepsilon \Rightarrow \varepsilon(t) = \varepsilon(0)e^{{-2t}}\), giving \(\mu_{{\max}} = \lim_{{t\to\infty}} \frac{{1}}{{t}}\ln\left|\frac{{\varepsilon(t)}}{{\varepsilon(0)}}\right| = -2\) exactly. Concretely: a perturbation loses 86.5% of its size after \(t=1\), 98.2% after \(t=2\), 99.8% after \(t=3\). The system forgets its initial condition fast.
By full nonlinear Gronwall estimate (not just the linearisation), \(|r(t)-1| \le \varepsilon_0 \cdot e^{{-2t}}\) holds uniformly with \(\varepsilon_0=1/3\) — proved, gronwall_outer, 0 sorry. Any trajectory starting within \(1/3\) of \(r=1\) is guaranteed to converge to \(\Gamma\). Week 3 covers what happens outside that guaranteed radius.
-- dm³ 102 · Week 02 · μ_max derivation
-- Toy system (contact form α = dz - r²dθ):
-- ṙ = r(1-r²) + 2(r-1)e^{-z}, θ̇ = 1, ż = r² - 2(r-1)²e^{-z}
-- Linearise at r=1: ε̇ = -2ε ⟹ μ_max = -2 (exact, not fit)
-- gronwall_outer : |r(t)-1| ≤ ε₀·e^{-2t}, ε₀=1/3 proved, 0 sorry
example : (1:ℝ)/3 = 2/(2*3) := by norm_num -- ε₀ = |μ_max|/(2(1+sup‖Hess V‖))