dm³ ODE numerical summary
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System:
  ṙ = r(1 - r²) + 2(r - 1) e^(-z)
  θ̇ = 1
  ż = r² - 2(r - 1)² e^(-z)

Equilibrium structure:
  Surface r = 1 is invariant (ṙ = 0 identically).
  On r = 1:  θ̇ = 1, ż = 1.  Trajectory is a helix with pitch 2π.

Linearization near r = 1:
  ṙ ≈ 2ε(e^(-z) - 1) for r = 1 + ε
  Attracting when z > 0, repelling when z < 0.
  Since ż = 1 on r = 1, trajectories escape to z → ∞ and stay attracting.

Empirical stability radius at z₀ = 0.5:  ε ≈ 0.375
  (1/3 ≈ 0.333, 1/6 ≈ 0.167, 2/3 ≈ 0.667)